Fields Institute Communications Volume 00, 0000

The theorem of Grunwald-Wang in the setting of valuation theory

Falko Lorenz

Mathematisches Institut Universität Münster (Germany) lorenz@math.uni-muenster.de

Peter Roquette

Mathematisches Institut Universität Heidelberg (Germany) roquette@uni-hd.de

Abstract. Given a field K with finitely many valuations; does there exist an extension of K which at these valuations has a prescribed local behavior? The Grunwald-Wang theorem answers this question in the case of abelian field extensions. Originally developed for algebraic number fields in the context of class field theory, it has turned out that it is valid quite generally, for arbitrary multi-valued fields, provided the valuations are of rank one or, more generally, are mutually independent and dense in their respective henselizations. In this paper we present a simple proof which is based on Kummer theory for cyclic Galois algebras, and on Witt theory in case of characteristic p.

Contents

1. Introduction 2 1.1. Statement of main results 2 1.2. Reduction to cyclic groups of prime power order 6 1.3. Plan of work 6 1.4. Further comments 7 2. Parametrization of cyclic Galois algebras 8 2.1. Kummer theory 8 2.2. Galois action on the n-th roots of unity 11 2.3. Parametrization in the non-exceptional case 12 2.4. Proof of Grunwald-Wang theorem 16 3. The exceptional case 17 3.1. Parametrization in the exceptional case 18 3.2. Proof of Grunwald-Wang theorem 24

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

1991 Mathematics Subject Classification. Primary 12J10; Secondary 11R37.

4. The case of characteristic p 24 4.1. Preliminaries on Witt vectors 24 4.2. Witt radicals for cyclic Galois algebras 25 4.3. Proof of Grunwald-Wang theorem 30 5. Appendix: On Galois algebras 32 References 39

1 Introduction

1.1 Statement of main results. A field K, equipped with a finite nonempty set V of inequivalent valuations, archimedean or non-archimedean, is called a multi-valued field. ^{1} Since we include the archimedean case we write the valuations multiplicatively; if a ∈ K and v ∈ V then |a|_{v} ∈ ℝ denotes the corresponding value, |a|_{v} ≥ 0. We put

|a|_{V} = max_{v∈V} |a|_{v}.

This defines a metric topology on K. Let K̂_{V} denote the corresponding completion. If it is clear from the context which set V we are referring to, then we omit the subscript V and write simply K̂. Since the valuations v ∈ V are independent, K̂ splits into the direct product of complete fields:

K̂ = ∏_{v∈V} K̂_{v} .  (1)

We see that K̂ is a commutative semisimple K-algebra. ^{2}

Now let L|K be a finite Galois extension with Galois group G. Every valuation v ∈ V has finitely many extensions to L, and so we obtain a finite set of valuations w of L. Thus L, as an extension of the multi-valued field K, is again multi-valued in a canonical way. We have

L̂ = ∏_{v∈V} L̂_{v}  with  L̂_{v} = ∏_{w|v} L̂_{w}  (2)

where w|v indicates that w is an extension of v. Note that L̂_{v} = L ⊗_{K} K̂_{v} since L|K is separable. Hence

L̂ = L ⊗_{K} K̂  (3)

The Galois group G acts on the left factor of this tensor product and hence on L̂. With respect to this action L̂ becomes a Galois G-algebra over K̂. ^{3}

The structure of L̂ as a Galois G-algebra over K̂ describes the local behavior of the valuations v ∈ V in the Galois extension L|K. For instance, the decomposition groups G_{v} ⊂ G of the valuations v ∈ V in the sense of valuation theory can be identified with the decomposition groups of the Galois G-algebra L̂|K̂. ^{4} There arises the question whether for a given finite group G, there exists a Galois extension L|K with Galois group G with prescribed local behavior at the valuations v ∈ V. In other words:

1In this paper all valuations are supposed to be of rank one, i.e., the values are real numbers. More generally, as the reader will observe, our arguments remain valid for finitely many independent Krull valuations of K such that K is dense in the respective henselizations.

2We use the terminology “semisimple algebra” in the sense that it implies the algebra to be Artinian. Hence a commutative semisimple algebra is a direct sum of finitely many fields, and conversely.

3See A.10. For the convenience of the reader, the definitions and basic facts concerning Galois algebras are recalled in the appendix. References to the appendix are prefaced by capital A. Thus A.10 refers to statement A.10 of the appendix.

Given a finite group G and a Galois G-algebra A over K̂, does there exist a Galois extension field L|K on which G acts as Galois group, such that L̂ is isomorphic to A as a Galois G-algebra over K̂?

L̂ ≈ A ? L G? L ||||||||| K K  K̂ GK̂ ||||||||| L̂ ≈ A

If G is an abelian group then the above question has been studied by W. Grunwald, a student of Hasse, in his 1933 paper [2], in the case when K is an algebraic number field of finite degree. He assumed that G is generated by the decomposition groups G_{v} for v ∈ V ^{5} and then claimed that, indeed, such L|K does exist. His proof was based on class field theory.

Grunwald’s theorem became important in the context of class field theory and the arithmetic theory of central simple algebras. In 1942 G. Whaples [13] presented a new proof which was based on class field theory too but did not use analytic methods, as it had been necessary in Grunwald’s time. ^{6}

However in 1948 Sh. Wang [10] presented a counter-example to Grunwald’s theorem, and 1950 in his thesis [11] he corrected the error in Grunwald’s (as well as in Whaples’) paper, giving precise conditions under which Grunwald’s theorem holds for an algebraic number field. It turned out that there are only certain “special” cases of number fields in which Grunwald’s theorem may fail to hold without further conditions, but that in all “non-special” cases the theorem holds as had been stated by Grunwald. Those “special” cases can occur only if the exponent of G is divisible by 8, and they depend on the behavior of the field of 2-power roots of unity over K.

Since then the theorem is called the Grunwald-Wang theorem.

In the same year 1950, H. Hasse [4] also presented a correction of Grunwald’s theorem in the context of class field theory; Hasse had known Wang’s counterexample but not his thesis [11]. See also Chap. X of the Artin-Tate notes [1] on class field theory for a discussion of the Grunwald-Wang theorem. Hasse asked whether a proof of the Grunwald-Wang theorem could be given by means of Kummer theory, independent of class field theory.

4For the notions and facts about decomposition groups of Galois G-algebras see the appendix, in particular A.6.

5In case of a number field Grunwald’s assumption is no essential restriction; see Corollary 3 below.

6Whaples erroneously called it “Gruenwald’s theorem” but the correct name is “Grunwald”.

Here we shall answer Hasse’s question and prove, on the basis of Kummer theory for Galois algebras, that the Grunwald-Wang theorem is valid for arbitrary multi-valued fields. (In case of characteristic p we use the additive theory of Witt vectors instead of the multiplicative Kummer theory.)

In this generality, however, the existence of a Galois field extension with group G cannot be expected (not for instance, if K is algebraically closed). Instead, we can only assert the existence of a Galois G-algebra L|K whose completion is isomorphic to A. The easiest way to define the completion L̂ of a Galois G-algebra L|K over a multi-valued field K is by using the formula (3); in the case of a multi-valued field extension this coincides with the definition given above. In any case, formula (3) is what we will refer to in our proof.

Before stating our main result we have to give the definition of “non-special”. Let n be an integer. Following Artin-Tate [1] we shall call a field “non-special” with respect to n if it satisfies the following

Wang condition: Let 2^{ν} denote the highest power of 2 dividing n; then the field of 2^{ν}-th roots of unity is cyclic over K. ^{7}

If K is of prime characteristic > 0 then the Wang condition is always satisfied, for every n. If char(K) = 0 and n is odd then, again, the Wang condition is satisfied; more generally this holds for ν ≤ 2. However, if ν ≥ 3 then, for instance, the rational number field ℚ does not satisfy the Wang condition. The field of 8-th roots of unity over ℚ is of degree 4 and generated by square roots:

ℚ(\sqrt[8]{1}) = ℚ(\sqrt{−1}, \sqrt{−2}) .

Its Galois group is not cyclic. This fact was essential in Wang’s counter-example: He worked with K = ℚ as base field, and with V consisting solely of the 2-adic valuation of ℚ. Wang showed, and this is not difficult, that there does not exist a cyclic extension L|ℚ of degree 8 such that L̂_{2}|ℚ̂_{2} is the unramified field extension of degree 8.

If a field K contains at least one of the square roots \sqrt{−1}, \sqrt{−2} then again, K satisfies the Wang condition, for all n.

For a detailed description of the special cases we refer to the literature, e.g., to Artin-Tate [1] Chap.X.

Our main result can now be stated as follows:

Theorem 1 (General Grunwald-Wang theorem) (i) Let (K, V) be a multi-valued field with completion K̂. Let G be a finite abelian group, of exponent n, and let a Galois G-algebra A over K̂ be given. Then there exists a Galois G-algebra L|K such that its completion L̂ is isomorphic to A as a Galois G-algebra over K̂ – provided K satisfies the Wang condition with respect to n.

(ii) It suffices already that every completion K̂_{v} for v ∈ V satisfies the Wang condition with respect to n.

From the definition we see that the Wang condition for K implies the Wang condition for every overfield of K, in particular for the completions K̂_{v} (v ∈ V).

7This should include the case char(K) = 2 when there are no proper roots of unity of 2-power order.

Hence the statement (ii) contains (i) but is more general. We have separated (ii) from (i) because this reflects our proof: First we shall prove Theorem 1 under the assumption that K satisfies the Wang condition and then we amend our proof such as to cover also the more general case (ii); this will be done in section 3. Note that if n is odd then the Wang condition is always satisfied, hence the odd case is covered by statement (i).

Remark: If a valuation v is non-archimedean of residue characteristic > 2 then its completion K̂_{v} satisfies the Wang condition; same for an archimedean valuation. Consequently, if V contains valuations v of this kind only, then (ii) shows that the conclusion of the Grunwald-Wang theorem does hold without mentioning any extra condition.

As said above already, the existence of a Galois field extension L|K cannot be expected in the general situation of Theorem 1. However, if we assume (like Grunwald has done) that the decomposition groups of A generate G then it turns out that, indeed, L|K is a Galois field extension. This is easy to see. For, as explained in the appendix, the Galois G-algebra L|K has only one decomposition group H ⊂ G since its base K is a field. From L ⊗_{K} K̂ ≈ A it follows that the decomposition groups of A are subgroups of H (see A.10). If these subgroups generate G then it follows H = G and, hence, L|K is a field. Thus we can state the following result as an addition to the general Grunwald-Wang theorem.

Theorem 2 (Irreducibility theorem) Consider the same situation as in Theorem 1. Suppose that the decomposition groups of the given Galois G-algebra A|K̂ generate the group G. Then the Galois G-algebra L|K as announced in that theorem is in fact a Galois field extension.

We have called this the “Irreducibility theorem” since a Galois algebra is said to be irreducible if it is not properly decomposable into a direct sum, i.e., if it is a Galois extension of fields.

Corollary 3 (Number field case) Suppose that K is a number field of finite degree. Then the Grunwald-Wang theorem can be sharpened, to the effect that the Galois G-algebra L|K as announced in that theorem, can be chosen to be a Galois field extension – regardless of whether the decomposition groups of A|K̂ generate G or not.

Proof : Suppose the decomposition groups of A|K̂ do not generate G. Then we choose a cyclic subgroup H ⊂ G which is not contained in the group generated by the decomposition groups of A. We enlarge the given set V of valuations of K by adding a non-archimedean valuation v of K, independent of the valuations in V. We take care that K̂_{v} satisfies the Wang condition, e.g., by choosing v such that the residue characteristic is ≠ 2. There exists a cyclic extension M_{v}|K̂_{v} of degree |H|, e.g., the unique unramified extension of degree |H|. There exists an isomorphism of the Galois group of M_{v}|K̂_{v} with H. In other words: we have an injection Gal(M_{v}|K̂_{v}) ↪ G with image H. This injection gives rise to a Galois G-algebra A_{v}|K̂_{v} by means of the induction process from H to G (see A.7). By construction, the decomposition group of A_{v}|K̂_{v} is H.

Now let V′ := V ∪ {v}. This is a finite set of valuations of K. Its completion is K̂_{V′} = K̂_{V} × K̂_{v}. If we put A′ = A × A_{v}, then A′ is a Galois G-algebra over K̂_{V′} and the set of decomposition groups of A′ contains H, besides of the decomposition groups of A. Thus the subgroup of G generated by the decomposition groups of A′ is larger than that with respect to A. Repeating this process we finally obtain a finite set of valuations V′ ⊃ V and a Galois G-algebra A′|K̂_{V′} whose decomposition groups generate G; moreover, A′ contains the given A as a direct factor. Applying Theorem 2 to V′ and A′ we obtain a Galois field extension L|K with group G such that L̂_{V′} ≈ A′. By construction, this implies L̂_{V} ≈ A.

Remark: The same argument works not only for a number field of finite degree, but for any field K which carries infinitely many independent valuations v whose completions K̂_{v} satisfy Wang’s condition and admit cyclic field extensions of a given degree m. For instance, K may be an algebraic function field of one or several variables over some subfield.

1.2 Reduction to cyclic groups of prime power order. If the group G is a direct product G = G_{1} × G_{2} then the Galois G-algebra A|K̂ is isomorphic to a tensor product A = A_{1} ⊗K̂ A_{2} of a Galois G_{1}-algebra A_{1}|K̂ with a Galois G_{2}-algebra A_{2}|K̂ (see A.11). It suffices to prove the Grunwald-Wang theorem for each of the factors A_{1}, A_{2}.

For, if there exists a Galois G_{1}-algebra L_{1} over K such that L̂_{1} ≈ A_{1}, and similarly L̂_{2} ≈ A_{2}, then L = L_{1} ⊗_{K} L_{2} is a Galois G-algebra over K (see A.11 again) and L̂ = L̂_{1} ⊗K̂ L̂_{2} ≈ A_{1} ⊗K̂ A_{2} = A. Observe that the exponent n of G is the least common multiple of the exponents n_{1}, n_{2} of G_{1} and G_{2}. Hence if K satisfies the Wang condition with respect to n then it does so for n_{1} and n_{2} too.

Accordingly, we assume from now on that G is not decomposable as a direct product. Since G is assumed to be abelian this implies that G is cyclic of prime power order. The order of G is denoted by

|G| = n = p^{ν} .

The Wang condition can now be formulated to say that the field K( \sqrt[n]{1}) of n-th roots of unity should be cyclic over K. As said above, this is always satisfied if p > 2, or if char(K) > 0, or if at least one of the square roots \sqrt{−1}, \sqrt{−2} is contained in K.

1.3 Plan of work. Our main idea of proof is to use a parametrization of Galois G-algebras for a cyclic group G of prime power order n = p^{ν}. If the given Galois G-algebra A|K̂ is described by parameters in K̂ then after a small perturbation of those parameters one can assume that they are contained in K already. But in K they define a Galois G-algebra over L|K which then turns out to be a solution of the Grunwald-Wang problem.

This simple idea works well in the case when p ≠ char(K) and the n-th roots of unity are contained in K. For in this case we can use Kummer theory of Galois G-algebras; these can be parametrized by their Kummer radicands. We have to use Kummer theory over commutative semisimple algebras and not only over fields; for the convenience of the reader we include a short presentation of Kummer theory in this framework. See section 2.1.

But if the n-th roots of unity are not in K then the situation becomes a little more involved. Our idea is first to adjoin the n-th roots of unity to K, and then characterize those Kummer radicands whose corresponding Galois G-algebra has been obtained by base extension. It is here where we have to impose the Wang condition with respect to the group order n = p^{ν}. If the Wang condition is not satisfied (and hence p = 2) then a parametrization of the Galois G-algebras is still possible but not in a form which allows the use of the perturbation idea mentioned above. In fact, Wang’s counter-example shows that this is not a failure of the method but that the theorem fails to hold in general.

But even if the Wang condition is satisfied there is a certain exception. Whereas “in general” the Galois G-algebras can be parametrized by just one parameter (which we call “Kummer parameter”, see Prop. 8 in section 2.3) there are certain exceptions where two parameters are needed. These exceptions arise when p = 2 and \sqrt{−1} ∉ K. Although in this exceptional case our main idea is still applicable, we have to treat both cases separately. Thus we first deal with the non-exceptional case in section 2.3. The exceptional case, where p = 2, is treated in section 3, together with the proof of statement (ii) of the theorem.

Once the basic facts on the Kummer parametrization of cyclic Galois algebras are available, the proof of the Grunwald-Wang theorem turns out to be quite short and straightforward (see sections 2.4 and 3.2).

Finally, there is the case p = char(K). This case can be treated without problems by using Witt’s parametrization of cyclic Galois G-algebras of p-power rank. Witt has developed this theory for cyclic field extensions only; hence we shall briefly treat Witt’s theory in the framework of Galois G-algebras. See section 4.

1.4 Further comments. (1) Connection to embedding problem: The larger part of our paper is concerned with the presentation of Kummer parametrization and, in the case of p = char(K), of Witt parametrization. We would like to point out that this may be regarded as “well known” in the sense, that it can be extracted from the general theory of embedding problem for Galois G-algebras with abelian groups G. This theory has been started from a systematic point of view by Hasse [3] in a series of three papers on this subject. In modern language, the parametrization of such algebras can be described by certain cohomology invariants. But it would have taken some space and effort to show that those cohomology invariants can be parametrized by elements in the base algebra, in the way which we need for our proof of the Grunwald-Wang theorem. Hence we have decided to give a direct and relatively short presentation of the material, in a form which is suitable for our purpose. In order to put the simplicity of our method into evidence, we have tried to assume not too many prerequisites from algebra or cohomology theory. We have in mind a reader with the knowledge from an algebra course, say, [6].

(2) Historical remarks: We would like to point out that Miki [8] in 1978 had been the first who followed Hasse’s suggestion and proved Grunwald’s theorem in the setting of valuation theory, using Kummer theory as we do. His methods appear to be similar to ours, but he discussed discrete valuations only. And he imposed quite strong conditions concerning the “special” case, which later were relaxed by Sueyoshi [12] in 1980 but they are still more restrictive than ours. We believe that our method of putting the theorem in the framework of Galois algebras instead of Galois field extensions is more adapted to the problem. In fact, it puts into evidence that the discreteness of the valuations is not needed at all.

A completely new idea was introduced into the subject by the interesting paper of Saltman [9]. He approached the problem of the Grunwald-Wang theorem by means of his theory of generic Galois polynomials. In fact, our work started after reading Saltman’s paper; although the theory of generic Galois polynomials is of interest there arose the question whether his ideas of specialization could be transformed into parametrization. Although we do not do it explicitly in our paper, the reader will notice that if the Kummer parameter of a Galois G-algebra is transcendental over R in a suitable sense, then every Galois G-algebra over R is a specialization of that transcendental algebra.

(3) Open problem: In this paper we deal with the non-special case only. Over a number field, however, the “special” case is also discussed in the framework of class field theory, giving necessary and sufficient conditions for a Galois G-algebra A|K̂ to be the completion of a Galois G-algebra L|K. Those conditions refer to the behavior of local norm residue symbols of the field of roots of unity of 2-power order, locally at the critical valuations. There arises the question whether conditions of such type can be given in terms of Kummer parameters or similar invariants, over an arbitary multi-valued field. We leave it as an open problem to find such conditions.

2 Parametrization of cyclic Galois algebras

As said above, G is now supposed to be a cyclic group of prime power order n = p^{ν}. Let K be a field. Until section 4 it is assumed that p ≠ char(K).

2.1 Kummer theory. In the present section we assume in addition that the n-th roots of unity are contained in K. Let μ_{n} ⊂ K^{×} denote the group of n-th roots of unity and

χ: G → μ_{n}  (4)

an isomorphism from G to μ_{n}. This isomorphism is kept fixed in the sequel and all statements refer to the given χ although this is not mentioned explicitly.

Let R be a semisimple commutative K-algebra. Our aim is to give a description of the Galois G-algebras over R.

R^{×} denotes the multiplicative group of units, i.e., invertible elements, in R. Let a ∈ R^{×}. Consider an R-algebra A_{a} = R[x] which is generated by an element x satisfying

x^{n} = a  (5)

as a defining relation over R. This means that any other polynomial relation for x over R is a consequence of the relation (5). In other words: Let R[X] denote the polynomial algebra over R and consider the map R[X] → A_{a} given by X ↦ x, then the kernel of this map should be the ideal generated by the polynomial X^{n} − a, so that we obtain an isomorphism

R[X]/(X^{n} − a) ≈ A_{a}  (6)

of R-algebras. If y ∈ A_{a} is any other element satisfying y^{n} = a then there is a unique R-algebra homomorphism A_{a} → A_{a} such that x maps onto y.

Let σ ∈ G. We have (χ(σ)x)^{n} = x^{n} = a; hence there is a unique R-algebra homomorphism σ: A_{a} → A_{a} which takes x into

x^{σ} = χ(σ)x  (σ ∈ G) .  (7)

The homomorphism property χ(στ) = χ(σ)χ(τ) yields x^{στ} = (x^{σ})^{τ}. In this way A_{a} becomes a G-algebra. By construction, it is uniquely determined by a (up to isomorphisms of G-algebras over R).

Proposition 4 (Kummer Theory for Galois algebras) (i) Let a ∈ R^{×}. Any G-algebra A|R generated by an element x ∈ A satisfying the relations (5),(7) is a Galois G-algebra over R, and A is isomorphic to A_{a}.

(ii) If a ≡ b mod R^{×n}, i.e., if a = b · u^{n} with u ∈ R^{×}, then A_{a} is isomorphic to A_{b}, and conversely.

(iii) Every Galois G-algebra A|R is isomorphic to A_{a} for suitable a ∈ R^{×}. Such element a is called a Kummer radicand of A, and the corresponding x is called a Kummer radical of A.

Proof of (i): First we show that A_{a} is a Galois G-algebra. We decompose R into a direct product of fields; using A.3 we see that it suffices to prove the assertion for each direct factor separately. In other words: we may assume that R is a field. We use A.4 and see that we have to prove the following:

(a) A_{a} is semisimple.

(b) Fix(G, A_{a}) = R.

(c) [A_{a} : R] = n.

Statement (a) follows from (6) since the polynomial X^{n} − a has no multiple roots. Note that a ≠ 0, and that n is not divisible by the characteristic of the field R.

Statement (c) follows also from (6) since the n elements 1, x, …, x^{n−1} form an R-basis of A_{a}.

As to statement (b), let u ∈ A_{a} and write

u = ∑_{0≤i≤n−1} c_{i}x^{i}  with  c_{i} ∈ R.

For σ ∈ G we have according to (7):

u^{σ} = ∑_{0≤i≤n−1} c_{i}χ(σ)^{i}x^{i} .

Hence if u is fixed under σ then by comparing coefficients we obtain c_{i} = c_{i}χ(σ^{i}). Now take σ to be a generator of the cyclic group G; then we have χ(σ^{i}) ≠ 1 for 1 ≤ i ≤ n − 1 since χ is an ismomorphism. It follows c_{i} = 0 for those i (since R is a field) and hence u = c_{0} ∈ R.

Now let A be an arbitrary G-algebra over R, generated by an element x satisfying (5) and (7). We do not require a priori that the relation (5) is a defining relation for x over R. In any case, we have a unique R-homomorphism R[X] → A mapping X to x ∈ A. This is surjective because A is generated by x. In view of (5) this factors through the residue class algebra modulo X^{n} − a, and we obtain a homomorphism R(X)/(X^{n} − a) → A as R-algebras, mapping the residue class of X to x. In view of (7) this is a homomorphism of G-algebras.

Now, we know from the above that A_{a} = R(X)/(X^{n} −a) is a Galois G-algebra. Hence the map A_{a} → A is injective; see A.12. Since it is surjetive by construction, it is an isomorphism. (And, hence, the relation (5) is indeed a defining relation for x over R.)

Proof of (ii): Consider the Galois G-algebra A_{a} = R[x] with (5) and (7), and similarly A_{b} = R[y]. Suppose that a = bu^{n} with u ∈ R^{×}. We are going to establish an isomorphism φ : A_{a} → A_{b} as G-algebras over R.

The relation x^{n} = a is a defining relation for x over R. Hence, in order to obtain an homomorphism φ : A_{a} → A_{b} as R-algebras, it is sufficient to assign to x some element z ∈ A_{b} such that z satisfies the same relation z^{n} = a as x does. Clearly this holds for z := yu (since bu^{n} = a).

Hence we have a uniquely defined homomorphism φ : A_{a} → A_{b} as R-algebras, such that x^{φ} = yu. We claim that this is a homomorphism as G-algebras, which means that σφ = φσ for σ ∈ G. Indeed: x^{σφ} = (χ(σ)x)^{φ} = χ(σ)x^{φ} = χ(σ)yu = y^{σ}u = (yu)^{σ} = x^{φσ} since σ acts trivially on u ∈ R.

Since both A_{a} and A_{b} are Galois G-algebras over R it follows that φ : A_{a} → A_{b} is an isomorphism. (See A.12.)

Conversely, assume that A_{a} ≈ A_{b}. Let us identify A_{a} = A_{b} = A by means of that isomorphism. Thus on the one hand, A = R[x] with the relations (5), (7), and on the other hand A = R[y] with corresponding relations for y and b. When applying σ ∈ G both x and y take the same factor, namely χ(σ). Let us put u := yx^{−1}; this is fixed under G and hence u ∈ R. Note that x is a unit in A since x^{n} is a unit in R; similarly we have y ∈ A^{×}. It follows u ∈ R^{×}. From y = xu we compute b = y^{n} = x^{n}u^{n} = au^{n} ≡ a mod R^{×n} .

Proof of (iii): Now consider an arbitrary Galois G-algebra A over R; we have to find x ∈ A^{×} and a ∈ R^{×} satisfying the relations (5), (7) and such that A is generated over R by x. Since A is a Galois G-algebra it admits a normal basis over R. Let u ∈ A generate such a normal basis. Starting from such u we consider the “Lagrange resolvent”

x := ∑_{τ∈G} χ(τ)^{−1}u^{τ} .

A straightforward computation shows that x^{σ} = χ(σ)x, for σ ∈ G. Let us put a := x^{n}. Then a^{σ} = x^{σn} = (χ(σ)x)^{n} = x^{n} = a . Hence a is fixed under G and therefore contained in R.

If a would not be a unit in R then there would exist a primitive idempotent e ≠ 0 in R such that ea = 0. (Observe that R is supposed to be semisimple.) It follows (ex)^{n} = e^{n}x^{n} = ea = 0 and hence ex = 0, since A is commutative and semi-simple and therefore has no nilpotent elements ≠ 0. We have 0 = ex = ∑_{σ∈G} eχ(σ^{−1})u^{σ}. Since the u^{σ} form an R-basis of A we conclude that eχ(σ^{−1}) = 0 for all σ ∈ G. Taking σ = 1 we obtain e = 0, a contradiction. Thus indeed a ∈ R^{×}. Since x^{n} = a we conclude x ∈ A^{×}.

Thus x ∈ A^{×} satisfies relations of the form (5), (7). From (i) we conclude that the algebra R[x] is a Galois G-algebra, with the action of G induced by its action on A. The inclusion map R[x] ↪ A is a homomorphism of Galois G-algebras over R, hence an isomorphism by A.12. Thus R[x] = A.

We can reformulate Proposition 4 as follows:

Every element a ∈ R^{×} defines (uniquely up to isomorphisms) a Galois Galgebra A|R such that a is a Kummer radicand of A. The structure of A depends only on the residue class of a modulo n-th powers in R^{×}. Conversely, every Galois G-algebra A|R admits an element a ∈ R^{×} as its Kummer radicand.

Remark: Let A|R be a Galois G-algebra and a ∈ R^{×} a Kummer radicand of A. The corresponding Kummer radical x ∈ A of a is not uniquely determined. An element y ∈ A is another Kummer radical of a if and only if y = ζx with ζ ∈ R and ζ^{n} = 1. If R is a direct product of d fields then there are n^{d} such elements ζ. The n^{d} substitutions x ↦ ζx yield n^{d} automorphisms of A as Galois G-algebra over R, and every automorphism of A is of this form.

2.2 Galois action on the n-th roots of unity. Now we drop the assumption that the n-th roots of unity are contained in K.

Let K′ = K( \sqrt[n]{1}) be the field of n-th roots of unity over K. As in (4) we fix an isomorphism χ of G onto the group μ_{n} ⊂ K^{′×} of n-th roots of unity Let 𝔤 denote the Galois group of K′|K. Every automorphism τ ∈ 𝔤 is uniquely determined by its action on μ_{n}. There exists t ∈ ℤ such that

χ(σ)^{τ} = χ(σ)^{t}  (σ ∈ G) .  (8)

The exponent t modulo n is uniquely determined by τ . The mapping τ ↦ t gives an injection of the Galois group 𝔤 into the group (ℤ/n)^{×}.

Let s denote the order of τ. We have

τ^{s} = 1  hence  t^{s} ≡ 1 mod n .

Let ℓ ∈ ℤ be defined as

ℓ = \frac{t^{s} − 1}{n} .  (9)

At this point we use the assumption introduced in section 1.2 that n = p^{ν} is a prime power, p ≠ char(K). We shall need it in the proof of the following lemma. ^{8}

Lemma 5 (Normalization) Let τ ∈ 𝔤. The exponent t ∈ ℤ can be normalized in its residue class modulo n such that gcd(ℓ, n) = 1 except in the case when p = 2, s = 2, t ≡ −1 mod n. In this exceptional case we normalize t = −1 hence ℓ = 0.

Proof : n = p^{ν} is a p-power. The condition gcd(ℓ, n) = 1 is satisfied if

t^{s} ≢ 1 mod p^{ν+1} .  (10)

Suppose that t^{s} ≡ 1 mod p^{ν+1}, then we try to replace t by t + p^{ν}. We have

(t + p^{ν})^{s} ≡ 1 + st^{s−1}p^{ν} mod p^{ν+1} .

If s ≢ 0 mod p then we see that t + p^{ν} satisfies (10). If however s ≡ 0 mod p then we argue as follows: t^{s/p} is of order p modulo p^{ν}. If p > 2 the group (ℤ/p^{ν})^{×} is cyclic and, hence, it admits only one subgroup of order p. This subgroup consists of the p elements 1 + ip^{ν−1} for 0 ≤ i ≤ p − 1. Hence

t^{s/p} ≡ 1 + ip^{ν−1} mod p^{ν}  for some  i ≢ 0 mod p .

Taking p-th powers we conclude that t^{s} ≡ 1 + ip^{ν} mod p^{ν+1} and hence (10).

Now let p = 2. Every element in (ℤ/2^{ν})^{×} has order a power of 2. If ν ≥ 3 then there are 3 elements of order 2 in (ℤ/2^{ν})^{×}, namely −1 and ±1 + 2^{ν−1}. If t^{s/2} ≡ ±1 + 2^{ν−1} mod 2^{ν} then after squaring we obtain t^{s} ≡ 1 + 2^{ν} mod 2^{ν+1}, thus again (10). The exceptional case t^{s/2} ≡ −1 mod 2^{ν} can occur only if s = 2 since −1 is not a square in (ℤ/2^{ν})^{×}.

In the following we assume that the exponent t is normalized in the above sense. We define τ to be non-exceptional if indeed gcd(ℓ, n) = 1, and otherwise exceptional. If τ is exceptional then ℓ = 0.

Perhaps it is not unnecessary to mention that the trivial automorphism τ = 1 is non-exceptional; in this case we normalize t to be t = 1 + n and hence ℓ = 1.

Definition. 𝔤 is called exceptional if it contains an exceptional automorphism.

8This is taken from Saltman [9].

If this is the case then necessarily p = 2 and there is only one exceptional automorphism in 𝔤, namely the automorphism of order s = 2 which keeps the elements in K fixed and maps every n-th root of unity into its inverse.

Lemma 6 If 𝔤 is non-exceptional then 𝔤 is cyclic, hence K satisfies the Wang condition with respect to n.

Proof : If 𝔤 is non-cyclic then necessarily p = 2 and 𝔤 is isomorphic to a subgroup of (ℤ/2^{ν})^{×}. The group (ℤ/2^{ν})^{×} is non-cyclic if ν ≥ 3, and then it is generated by −1 (of order 2) and 5 (of order 2^{ν−2}):

(ℤ/2^{ν})^{×} = ⟨−1⟩ × ⟨5⟩ .  (11)

Every non-cyclic subgroup of (ℤ/2^{ν})^{×} necessarily contains −1, and this corresponds to the exceptional automorphism of 𝔤.

Remark: The invariants s, t, ℓ describe the action of τ ∈ 𝔤 on the n-th roots of unity. If we wish to indicate which automorphism τ they belong to then we write more precisely s_{τ} , t_{τ} , ℓ_{τ}. But for simplicity of notation we mostly drop the index τ if it is clear from the context which automorphism τ we are referring to.

In addition we will have to use the operator N = N_{τ} as follows:

N = t^{s−1} + t^{s−2}τ + ⋯ + tτ^{s−2} + τ^{s−1} .  (12)

N operates on any module on which τ operates. We have (t−τ)N = t^{s} −τ^{s} = t^{s} −1 and hence, using the definition (9) of ℓ:

(t − τ)N = ℓn .  (13)

We will use this relation several times in the sequel. ^{9}

2.3 Parametrization in the non-exceptional case. Let R be a commutative semisimple K-algebra. As above, K′ denotes the field of n-th roots of unity over K. We put

R′ = K′ ⊗_{K} R  (14)

and call R′ the “algebra of n-th roots of unity” over R. The Galois group 𝔤 of K′|K acts on the left hand side of the tensor product, and by this action R′ becomes a Galois 𝔤-algebra over R (see A.10). On the other hand, R′ can be regarded as a commutative semisimple algebra over K′, and since K′ contains the n-th roots of unity, the Kummer Theory of Proposition 4 can be applied to Galois G-algebras over R′.

9The notation N for the operator on the right hand side of (12) has been chosen to indicate that it behaves somewhat like the norm operator in cohomology theory. In fact, if we introduce the so-called “twist” \overline{τ} = tτ^{−1} and its norm \overline{N} = \overline{τ}^{s−1} + \overline{τ}^{s−2} + ⋯ + 1 then N = τ^{(s−1)}\overline{N}. By using the twist \overline{τ} instead of τ it would have been possible to put our following discussion into a more systematic and general framework of cohomology theory. For, the condition of Prop. 7 below can be stated to the effect that a′, if regarded in the factor group R^{′×}/R^{′×n}, should be a norm with respect to the twist \overline{τ}, which can be interpreted that the comology class determined by a′ in H^{0}(\overline{𝔤}, R^{′×}/R^{′×n}) is trivial. In this form, Prop.7 can be regarded as a very special case of Hasse’s general theory of abelian algebras [3], as we have already mentioned in the introduction. (However, Hasse’s theory would first have to be extended, such as to cover not only abelian algebras over a field but also over a semisimple algebra as a base.) After some deliberation we have decided not to work with the twist (which nowadays is also called “Tate twist”) because it seems not to be relevant for our purpose, namely the parametrization of Galois G-algebras for cyclic G.

Consider a Galois G-algebra A over R and put

A′ = A ⊗_{R} R′ .  (15)

Then A′ is a Galois G-algebra over R′ (see A.10). It is obtained from A by extending the base algebra from R to R′. Let a′ ∈ R^{′×} be a Kummer radicand of A′. We ask:

What are the properties of a′ which express the fact that A′ is obtained from the Galois G-algebra A over R by base extension from R to R′?

𝔤 A ⊗_{R} R′ = A′ ? A G? A |||||||||R  𝔤 R  R′ GR′ ||||||||| A ⊗_{R} R′ = A′

In this section we treat the case that 𝔤 is non-exceptional in the sense as defined in section 2.2. By Lemma 6 this implies that 𝔤 is cyclic. Let us choose a generator τ ∈ 𝔤, so that

𝔤 = ⟨τ⟩ .

τ is a non-exceptional automorphism in 𝔤. The invariants s, t, ℓ, N belonging to τ have been introduced in section 2.2.

Proposition 7 Suppose that 𝔤 = ⟨τ⟩ is non-exceptional. Let A′|R′ be a Galois G-algebra with Kummer radicand a′ ∈ R^{′×}. If there exists a ∈ R^{′×} such that a′ ≡ a^{N} mod R^{′×n} then A′ is representable as a tensor product A′ = A ⊗_{R} R′ with some Galois G-algebra A|R. And conversely.

This is immediate from the following result which gives a complete parametrization of the Galois G-algebras A|R.

Proposition 8 (Parametrization, non-exceptional case) Suppose that 𝔤 is non-exceptional and that 𝔤 = ⟨τ⟩. Then:

(i) Every element a ∈ R^{′×} determines a Galois G-algebra A(a) over R by way of the following two-step construction.

Step 1:: Construct the Galois G-algebra A′ over R′ with a^{N} as its Kummer

radicand. Hence A′ = R′[x] with x^{n} = a^{N} and x^{σ−1} = χ(σ). Step 2:: Extend the action of 𝔤 on R′ to an action of 𝔤 on A′ such that x^{t−τ} = a^{ℓ} . This extension is possible and unique. Then put A(a) := Fix(𝔤, A′) ; this is a Galois G-algebra over R. Moreover, A(a) ⊗_{R} R′ ≈ A′.

(ii) If a ≡ b mod R^{′×n} then A(a) ≈ A(b).

(iii) Every Galois G-algebra A over R is isomorphic to A(a) for suitable a ∈ R^{′×}. Such an element a is called a Kummer parameter for A.

Remark: We can reformulate Proposition 8 as follows:

If 𝔤 is non-exceptional then the elements a ∈ R^{′×} parametrize the Galois Galgebras A|R (up to isomorphisms) as their Kummer parameters. Elements which differ by n-th power factors parametrize isomorphic Galois G-algebras.

Note that we do not claim the Kummer parameter a to be uniquely determined modulo n-th powers. It may well be that A(a) ≈ A(b) but a ≢ b mod R^{′×n}. If however K contains the n-th roots of unity then K′ = K and 𝔤 = 1, and the Kummer parameters coincide with the Kummer radicands, i.e., A(a) = A_{a} in the sense of Proposition 4.

Proof of (i): Let A′ be constructed as in Step 1. We try to extend the action of τ on R′ to an action on A′.

We recall from Kummer Theory (Prop. 4) that the relation x^{n} = a^{N} is a defining relation for x over R′. Consequently, in order to extend the action of τ to an algebra homomorphism of A′ it suffices to assign to x an element y ∈ A′ which satisfies the image of that relation under τ, which is to say y^{n} = a^{Nτ}. We claim that this holds for y = x^{t}a^{−ℓ}. Indeed, we compute using (13): (x^{t}a^{−ℓ})^{n} = a^{Nt}a^{−ℓn} = a^{Nt−ℓn} = a^{Nτ} . We conclude:

There is a unique algebra homomorphism τ of A′ into itself, extending the given algebra automorphism τ of R′ and satisfying

x^{τ} = x^{t}a^{−ℓ}  or, equivalently  x^{t−τ} = a^{ℓ} .  (16)

Next we claim that τ^{s} = 1, i.e., τ has order s not only as an operator on R′ but also on A′. Since this relation holds on R′ it suffices to verify it when applied to the generator x of A′|R′. Now, as operators on the unit group A^{′×} we have τ^{s} − 1 = (t^{s} − 1) − (t^{s} − τ^{s}) = ℓn − (t − τ)N; remember the definitions (9) of ℓ and (12) of N. Thus τ^{s} = 1 is equivalent to (t − τ)N = ℓn. We have to verify this when applied to x. Indeed, using (16) we have: x^{(t−τ)N} = a^{ℓN}, and on the other hand: x^{nℓ} = a^{Nℓ} since x^{n} = a^{N}.

Next we show that the action of τ on A′ commutes with the action of G, i.e. that στ = τσ holds on A′. Again, this relation holds on R′ (because σ acts trivially on R′) and therefore it suffices to verify this when applied to x. Indeed: x^{σ(t−τ)} = (χ(σ)x)^{t−τ} = x^{t−τ} = a^{ℓ} since τ acts on χ(σ) as the exponentiation with t. On the other hand, x^{(t−τ)σ} = a^{ℓσ} = a^{ℓ} since σ acts trivially on a^{ℓ} ∈ R′. Thus x^{σ(t−τ)} = x^{(t−τ)σ} and therefore x^{στ} = x^{τσ}.

We have proved the following

Statement (i):^{10} The relation (16) establishes uniquely an action of the cyclic group ⟨τ⟩ as a group of G-algebra automorphisms of A′, extending the action of ⟨τ⟩ on R′.

Now, according to the hypothesis of the Proposition, ⟨τ⟩ = 𝔤 and hence A′ appears as a 𝔤-algebra. In fact, A′ is a G × 𝔤-algebra since the actions of G and of 𝔤 on A′ commute elementwise. From the general theory we conclude that A′|R is in fact a Galois G × 𝔤-algebra (since R′|R is a Galois 𝔤-algebra and A′|R′ is a Galois G-algebra; see A.13). From this the other assertions of (i) follow, namely: the fixed algebra A(a) = Fix(𝔤, A′) is a Galois G-algebra over R, and A(a) ⊗_{R} R′ is isomorphic to A′ (see A.11).

10For later reference we observe that, in this part of proof, until now we have not used the fact that 𝔤 = ⟨τ⟩. Hence this statement remains valid for an arbitrary cyclic subgroup ⟨τ⟩ of 𝔤.

The group 𝔤 may be exceptional or not.

Proof of (ii): As above, A′ denotes the Galois G-algebra over R′ with Kummer radicand a^{N}; thus A′ = R′[x] with x^{n} = a^{N}. Similarly let B′ = R′[y] with y^{n} = b^{N}. Suppose that a = bu^{n} with u ∈ R^{′×}. We are going to establish an isomorphism φ : A′ → B′.

We have shown in part (ii) of the proof of Prop. 4 that the assignment

x^{φ} = yu^{N}  (17)

defines uniquely a G-isomorphism A′ → B′ over R′. Now we claim that, in addition, φ is compatible with the action of τ, i.e., that τφ = φτ. This holds on R′ since φ leaves the elements of R′ fixed. Hence again, it is sufficient to verify this when applied to x. Now from (16) we see that x^{(t−τ)φ} = a^{ℓφ} = a^{ℓ}; and on the other hand: x^{φ(t−τ)} = (yu^{N})^{t−τ} = b^{ℓ}u^{N(t−τ)} = b^{ℓ}u^{ℓn} = a^{ℓ}, where we have used (13). Thus x^{(t−τ)φ} = x^{φ(t−τ)} and therefore x^{τφ} = x^{φτ}. We have now shown the validity of

Statement (ii):^{10} Suppose a = bu^{n} with u ∈ R^{′×}. Then the relation (17) establishes an R′-algebra isomorphism φ : A′ → B′ which is compatible with the action of G and of τ.

Now, according to the hypothesis of the Proposition, τ generates 𝔤. Hence φ : A′ → B′ is an isomorphism of G-algebras and of 𝔤-algebras, i.e., of G × 𝔤-algebras. It follows that φ maps the fixed algebra A(a) = Fix(𝔤, A′) onto A(b) = Fix(𝔤, B′).

Proof of (iii): Now let A|R be an arbitrary Galois G-algebra. Let us put A′ := A ⊗_{R} R′. This is a Galois G × 𝔤-algebra and we have A = Fix(𝔤, A′) (see A.11). In particular, τ acts on A′, and this action commutes with every σ ∈ G. We are going to construct a Kummer radical x of A′|R′ such that, firstly, x^{n} = a^{N} with a ∈ A^{′×}, and secondly the relation (16) holds: x^{t−τ} = a^{ℓ}.

We start with an arbitrary Kummer radical x ∈ A^{′×}. Since τ commutes with σ ∈ G we have x^{(t−τ)σ} = x^{σ(t−τ)} = χ(σ)^{t−τ} x^{t−τ} = x^{t−τ}. Hence the element u := x^{t−τ} is kept fixed by all σ ∈ G and therefore u ∈ R^{′×}.

Now we use the fact that τ is non-exceptional, which by definition means that ℓ is relatively prime to n. Let m ∈ ℤ be a multiple of ℓ such that m ≡ 1 mod n. Then x^{m−1} ∈ R^{′×} since x^{n} ∈ R^{′×} and n|m − 1. Hence x^{m} = x · x^{m−1} differs from x by a factor from R^{′×}, and therefore x^{m} too is a Kummer radical of A′|R′. We put a = u^{m/ℓ} and compute: (x^{m})^{(t−τ)} = u^{m} = a^{ℓ}, and (x^{m})^{n} = x^{ℓn(m/ℓ)} = x^{(t−τ)N·(m/ℓ)} = u^{N·(m/ℓ)} = a^{N}, where we have used (13). We have shown:

Statement (iii):^{10} For any Kummer radical x of A′|R′ the element u := x^{t−τ} is contained in R^{′×}. Let m ∈ ℤ be a multiple of ℓ such that m ≡ 1 mod n (such m exists since τ is assumed to be non-exceptional). Then the m-th power x^{m} is also a Kummer radical of A′|R′. Its radicand is (x^{m})^{n} = a^{N} where a := u^{m/ℓ}. Moreover, we have (x^{m})^{(t−τ)} = a^{ℓ}.

Now, changing notation we write again x instead of x^{m}. We have found a Kummer radical x of A′|R′ such that x^{n} = a^{N} and x^{t−τ} = a^{ℓ} for some a ∈ R^{′×}, as required.

We shall have occasion to consider base extensions. Suppose that R is contained in the semisimple commutative K-algebra S. We assume that the unit element of R is also the unit element of S. Let A|R be a Galois G-algebra, and consider the Galois G-algebra A_{S} := A ⊗_{R} S over S. ^{11} We say that A_{S} is obtained from A by base extension from R to S.

Let a be a Kummer parameter for A, so that A = A(a). We claim that a is also a Kummer parameter for A_{S}. To see this, consider the diagram of R-algebras, with A′ = A ⊗_{R} R′:

𝔤  A′ A GA ||||||R  𝔤 R  R′ GR′ |||||| A′

Within A′, the algebra A is characterized as the fixed algebra A = Fix(𝔤, A′). By tensoring with S we obtain the diagram

𝔤  A′_{S} A_{S} GA_{S} |||||| R_{S} 𝔤 R_{S}  R′_{S} GR′_{S} |||||| A′_{S}

where A′_{S} = A′ ⊗_{R} S and similarly A_{S}, R_{S}, R′_{S}. Again, we have A_{S} = Fix(𝔤, A′_{S}) (see A.10). The action of 𝔤 = ⟨τ⟩ on A′ is given by the formulas

x^{n} = a^{N}  (18)
x^{σ−1} = χ(σ) (for σ ∈ G)  (19)
x^{t−τ} = a^{ℓ}  (20)

where A′ = R′[x]; this establishes a as a Kummer parameter of A. Now, the same formulas hold in A′_{S} = R′_{S}[x], and hence a is also a Kummer parameter of A_{S}. Thus we have:

Lemma 9 (Base extension) Let A be a Galois G-algebra over R with Kummer parameter a ∈ R^{′×}. If a is considered as an element in S^{′×}, then it is a Kummer parameter for the Galois G-algebra A_{S} = A ⊗_{R} S over S, which is obtained from A by base extension from R to S.

2.4 Proof of Grunwald-Wang theorem. Based on the above parametrization we will now give a simple proof of the Grunwald-Wang theorem. This proof is restricted to the non-exceptional case since we shall use Kummer parameters according to Proposition 8. The exceptional case will be treated in the next section.

11See A.10 for the fact that A ⊗_{R} S is a Galois G-algebra over S.

We consider the situation of the Grunwald-Wang theorem; thus K is a multivalued field and K̂ its completion. We have to use the following Lemma. Of course this is well known; we present the proof for the convenience of the reader.

Lemma 10 Suppose that n is not divisible by the characteristic of K̂. If z ∈ K̂ is sufficiently close to 1 then z = u^{n} is an n-th power, with u ∈ K̂^{×}.

Proof : K̂ is a direct product of the complete fields K̂_{v}. It suffices to discuss each factor K̂_{v} separately. In other words: we may assume that K̂ is a field with a single valuation |·|.

If the valuation of K̂ is archimedean, then either K̂ = ℝ or K̂ = ℂ. In this case the assertion is clear.

If K̂ is non-archimedean then we use the so-called Hensel-Rychlik Lemma. ^{12} Consider the polynomial f(X) = X^{n} − z; the condition u^{n} = z is equivalent to f(u) = 0. We have

|f(1)| = |z − 1|  and  |f′(1)| = |n| ≤ 1.

Note that |n| > 0 since n is not divisible by the characteristic of K̂. Therefore, if

|z − 1| < |n|^{2}  (21)

then the Hensel-Rychlik Lemma guarantees the existence of u ∈ K̂ such that

f(u) = 0  and  |u − 1| < |n| .

Proof of Grunwald-Wang theorem:

Besides of K we consider K′ = K( \sqrt[n]{1}) and its completion K̂′ = K′ ⊗_{K} K̂. We apply Prop. 8 to Galois G-algebras over R = K̂.

Let A be a Galois G-algebra over K̂, and let a ∈ K̂^{′×} be a Kummer parameter for A, according to Proposition 8. We observe that K′ is dense in K̂′; hence there are elements b ∈ K′ which are arbitrarily close to a. Then a^{−1}b is close to 1 and we infer from Lemma 10 (applied to K̂′ instead of K̂) that a^{−1}b ≡ 1 mod K̂^{′×n}. Hence by Proposition 8, b is also a Kummer parameter for A. We change notation and write again a instead of b. We have seen:

There exists a Kummer parameter a for A|K̂ which is contained in K′.

Now we use again Proposition 8, but over K instead of K̂. Hence a is the Kummer parameter of a certain Galois G-algebra L over K. We apply Lemma 9 with respect to the base extension K ⊂ K̂ and conclude that L ⊗_{K} K̂ ≈ A. In view of (3) we see that L̂ ≈ A.

3 The exceptional case

In this section we consider the case when 𝔤 is exceptional in the sense as defined in connection with Lemma 5. Then we have necessarily p = 2 and n = 2^{ν} ≥ 2. The group 𝔤 contains a unique exceptional automorphism; this will be denoted by ε. By definition, ε acts on the roots of unity as the inverse operator: χ(σ)^{ε} = χ(σ)^{−1}.

𝔤 is the direct product of the subgroup ⟨ε⟩ of order 2 and a cyclic group generated by a non-exceptional automorphism τ:

𝔤 = ⟨τ⟩ × ⟨ε⟩ .  (22)

12For this we refer to [6] II, §23, F14.

This follows from the fact that 𝔤 is isomorphic to a subgroup of (ℤ/2^{ν})^{×} which has the product decomposition (11), and ε corresponds to −1 in this isomorphism.

We do not exclude the case τ = 1 in which case 𝔤 = ⟨ε⟩ is cyclic and hence K satisfies the Wang condition with respect to n.

Let s denote the order of τ. The defining relations for τ and ε as generators of 𝔤 are

τ^{s} = 1 , ε^{2} = 1 , τε = ετ . (23)

The invariants s_{τ} , t_{τ} , ℓ_{τ} , N_{τ} referring to τ will simply be denoted by s, t, ℓ, N, as in the foregoing section. The corresponding invariants for ε are

s_{ε} = 2, t_{ε} = −1, ℓ_{ε} = 0, N_{ε} = ε − 1 .

We shall denote by K′_{ε} the fixed field of ε within K′. Thus K′|K′_{ε} is a quadratic field extension.

3.1 Parametrization in the exceptional case. Again, let R be a commutative semisimple K-algebra, and consider the Galois 𝔤-algebra R′ = K′ ⊗_{K} R of n-th roots of unity over R. Let

R′_{ε} = Fix(ε, R′) = K′_{ε} ⊗_{K} R

denote the subalgebra of the elements which are fixed under ε. Then R′ is a Galois ⟨ε⟩-algebra over R′_{ε} and R′_{ε} is a Galois ⟨τ⟩-algebra over R; see A.9. The map z ↦ z^{ε+1} is the ordinary norm map from the algebra R′ to R′_{ε}.

In the exceptional case it will turn out that not every a ∈ R^{′×} is admissible as a Kummer parameter for Galois G-algebras.

Definition: An element a ∈ R^{′×} is called admissible if its ε-norm a^{ε+1} is an n-th power in R_{ε}^{′×}, i.e., if there exists c ∈ R^{′×} such that

a^{ε+1} = c^{n}  and  c^{ε−1} = 1 .  (24)

If this is the case then (a, c) is called an admissible pair. The admissible pairs form a multiplicative subgroup W ⊂ R^{′×} × R_{ε}^{′×}.

In the exceptional case, the following propositions are analogous to Propositions 7 and 8.

Proposition 11 Suppose 𝔤 = ⟨τ⟩×⟨ε⟩ is exceptional. Let A′|R′ be a Galois G-algebra and a′ ∈ R^{′×} a Kummer radicand of A′. If there exists an admissible a ∈ R^{′×} with a′ ≡ a^{N} mod R^{′×n} then there exists a Galois G-algebra A over R such that A′ = A ⊗_{R} R′. The converse does also hold – provided the decomposition groups of the Galois 𝔤-algebra R′|R are cyclic.

This is an immediate consequence of

Proposition 12 (Parametrization; exceptional case) Suppose 𝔤 is exceptional, and write 𝔤 = ⟨τ⟩ × ⟨ε⟩ as explained above.

(i) Every admissible pair (a, c) ∈ W determines a Galois G-algebra A(a, c) over R by means of the following two-step construction.

Step 1:: Construct the Galois G-algebra A′|R′ with the Kummer radicand a^{N}.

Hence A′ = R′[x] with x^{n} = a^{N}and x^{σ−1} = χ(σ). Step 2:: Extend the action of 𝔤 on R′ to an action of 𝔤 on A′ such that x^{t−τ} = a^{ℓ} and x^{ε+1} = c^{N}. This extension is possible and unique. Then put A(a, c) := Fix(𝔤, A′) . This is a Galois G-algebra over R and A(a, c)⊗_{R} R′ ≈ A′.

(ii) If (b, d) is another admissible pair and (a, c) ≡ (b, d) mod W^{n} then A(a, c) ≈ A(b, d).

(iii) Suppose that the decomposition groups of the Galois 𝔤-algebra R′|R are cyclic. Then every Galois G-algebra A over R is isomorphic to A(a, c) for a suitable admissible pair (a, c) ∈ W. Such a pair is called a Kummer parameter pair for A.

Remark: We can reformulate Prop.8 as follows:

Let 𝔤 be exceptional. Suppose the decomposition groups of R′|R are cyclic. Then the admissible pairs (a, c) ∈ W parametrize the Galois G-algebras A|R (up to isomorphisms). Pairs which differ by n-th power factors from W parametrize isomorphic Galois G-algebras.

The proof of parts (i),(ii) will be quite analogous to the proof in the nonexceptional case, the only difference being that now, besides of τ, also the exceptional automorphism ε has to be considered. In part (iii) there will be some new consideration necessary, taking into account the hypothesis about the decomposition groups of R′|R.

Proof of (i): Let A′ be constructed as in Step 1. We try to extend the actions of both τ and ε on R′ to actions as G-algebra automorphisms of A′.

As to τ, we have done this in the proof of Prop. 8 already; see Statement (i) there. (Observe footnote 10.) Accordingly, the action of ⟨τ⟩ on R′ extends uniquely to an action of ⟨τ⟩ on A′ such that x^{t−τ} = a^{ℓ}.

As to ε, we argue as follows. In order to extend the action of ε to an algebra homomorphism of A′ it suffices to assign to x some element z ∈ A′ which satisfies the relations z^{n} = a^{Nε}. This holds for z = x^{−1}c^{N}. Indeed: we compute (x^{−1}c^{N})^{n} = a^{−N}c^{nN} = (a^{−1}c^{n})^{N} = a^{εN} where we have used the relation (24) which expresses admissibility of the pair (a, c).

As operator on A′, ε remains to be of order 2, for: x^{ε^{2}−1} = x^{(ε+1)(ε−1)} = c^{N(ε−1)} = c^{(ε−1)N} = 1 since c is fixed by ε in view of (24). Note that as operators on R′ we have τε = ετ and hence Nε = εN.

But we also have τε = ετ on A′. To see this we compute: x^{(t−τ)(ε+1)} = a^{ℓ(ε+1)} = c^{ℓn} in view of (24), and on the other hand x^{(ε+1)(t−τ)} = c^{N(t−τ)} = c^{ℓn} where we have used (13).

We have shown that the extended actions of τ, ε satisfy the defining relations (23) of the group 𝔤. In other words: The formulas

x^{t−τ} = a^{ℓ}  and  x^{ε+1} = c^{N}  (25)

define uniquely an extension of the action of 𝔤 = ⟨τ⟩ × ⟨ε⟩ on R′ to an action of 𝔤 as R-algebra automorphisms on A′.

Next we claim that 𝔤 acts on A′ by G-automorphisms, which is to say that τ as well as ε commute with each σ ∈ G. As to τ, we again refer to Statement (i) in the proof of Prop. 8. For ε we verify: x^{σ(ε+1)} = (χ(σ)x)^{ε+1} = x^{ε+1} = c^{N} since ε acts on χ(σ) as the exponentiation with −1. On the other hand, x^{(ε+1)σ} = c^{Nσ} = c^{N} since σ leaves c^{N} ∈ R′ fixed.

We have seen:

The action of 𝔤 on A′ given by (25) commutes elementwise with the action of G.

Hence A′ now becomes a G × 𝔤-algebra. Again, as in the non-exceptional case, we conclude that A′ is a Galois G×𝔤-algebra over R (since R′|R is a Galois 𝔤algebra and A′|R′ a Galois G-algebra; see A.13). From this it follows that A(a, c) = Fix(𝔤, A′) is a Galois G-algebra over R and that A(a, c) ⊗_{R} R′ ≈ A′ (see A.11).

Proof of (ii): As in the proof of Prop. 8(ii) we let A′ = R′[x] with x^{n} = a^{N} and B′ = R′[y] with y^{n} = b^{N}. Again, we have to exhibit an R′-isomorphism φ : A′ → B′ which is an isomorphism as G × 𝔤-algebras. Suppose that a = bu^{n} and c = dv^{n} with (u, v) ∈ W. Then we define φ by the same formula (17) as in the proof of prop. 8. According to Statement (ii) there, this indeed defines an R′- algebra isomorphism, and it is compatible with the action of τ and of each σ ∈ G (see footnote 10). It remains to verify that it is also compatible with ε. To this end we compute: x^{(ε+1)φ} = c^{Nφ} = c^{N} since φ acts as the identity operator on R′; on the other hand x^{φ(ε+1)} = (yu^{N})^{ε+1} = d^{N}u^{(ε+1)N} = (dv^{n})^{N} = c^{N} where we have used that the pair (u, v) is admissible and hence u^{ε+1} = v^{n}.

Proof of (iii): In this part of proof we suppose that the decomposition groups of R′|R are cyclic.

R is the direct product of the fields eR where e ranges over the set P (R) of primitive idempotents of R. For each e we have eR′ = K′ ⊗_{K} eR, thus we see that eR′ is the algebra of roots of unity over eR. If A|R is a Galois G-algebra then eA|eR is a Galois G-algebra too (see A.3). Accordingly it suffices for each e to exhibit an admissible pair in eW with respect to eA; since W is the direct product of the eW we obtain an admissible pair for A.

Consequently, we may assume from now on that R is a field.

Therefore there is only one decomposition group of R′|R, say 𝔥 (see A.6). 𝔥 is a subgroup of 𝔤, and 𝔥 is cyclic by hypothesis. In our discussion we will have to distinguish the cases ε ∉ 𝔥 and ε ∈ 𝔥. In each of these cases, given a Galois G-algebra A|R we have to construct a Kummer radical x of A′ := A ⊗_{R} R′ such that x^{n} = a^{N} for some admissible a ∈ R^{′×}, and that in addition (25) holds for some c ∈ R^{′×} for which (a, c) is an admissible pair. Note that A′ is a Galois G×𝔤-algebra, and hence τ, ε act on A′ as G-algebra automorphisms.

Case 1: ε ∉ 𝔥.

We first claim: If u ∈ R^{′×} is fixed under ε then there exists v ∈ R^{′×} such that u = v^{ε+1}. In other words: The norm map v ↦ v^{ε+1} from R^{′×} to R_{ε}^{′×} is surjective.

Proof : Consider the set P (R′) of primitive idempotents of R′. Since R is a field, 𝔤 acts transitively on P (R′). (See A.5.) The decomposition group 𝔥 is defined to be the stabilizer of an idempotent in P (R′). This does not depend on the choice of this idempotent since 𝔤 is abelian. Consequently, the assumption that ε ∉ 𝔥 implies that ε does not leave any primitive idempotent in P (R′) fixed. Accordingly P (R′) splits into pairs of idempotents which are mutually conjugate under ε. Let E be a set of representatives of those pairs, so that

P (R′) = E ∪ E^{ε} , E ∩ E^{ε} = ∅ .

Let e denote the sum of the primitive idempotents in E. Then e is an idempotent of R′ (not primitive in general) and we have

1 = e + e^{ε} , e · e^{ε} = 0 .

This leads to a direct product decomposition

R′ = eR′ × e^{ε}R′
= eR′ × (eR′)^{ε}
≈ S′ × S′  where S′ := eR′ .  (26)

Using this isomorphism, every u ∈ R^{′×} can be written as a vector in the form u = (u_{1}, u_{2}) with u_{1}, u_{2} ∈ S^{′×}. We have e = (1, 0) and e^{ε} = (0, 1). The automorphism ε acts on vectors by interchanging the components:

(u_{1}, u_{2})^{ε} = (u_{2}, u_{1}) .

If u^{ε} = u then u_{1} = u_{2}. Hence, putting v := (1, u_{1}) we have v^{ε+1} = (u_{1}, 1)(1, u_{1}) = (u_{1}, u_{1}) = u.

Our claim is proved.

Now let A|R be any Galois G-algebra, and A′ = A ⊗_{R} R′. We are looking for a Kummer radical x of A′|R′ such that x^{n} = a^{N}, and that (25) holds for some admissible pair (a, c).

We start with an arbitrary Kummer radical x ∈ A^{′×}. Since ε commutes with σ ∈ G we compute

x^{(ε+1)σ} = x^{σ(ε+1)} = χ(σ)^{ε+1}x^{ε+1} = x^{ε+1} ,  (27)

because ε acts on χ(σ) as the inverse operator. Thus x^{ε+1} is stable under G and therefore x^{ε+1} =: u ∈ R^{′×}. It follows u^{ε−1} = x^{ε^{2}−1} = 1 and therefore, as shown above, u = v^{ε+1} with some v ∈ R^{′×}. Hence (xv^{−1})^{ε+1} = 1. Changing notation and writing again x instead of xv^{−1} we have shown:

There exists a Kummer radical x of A′|R′ such that x^{ε+1} = 1.

With this Kummer radical x we now use the Statement (iii) as formulated in the proof of Prop. 8 (see footnote 10). We conclude that x^{m} is another Kummer radical of A′|R′, and that for certain a ∈ R^{′×} we have (x^{m})^{n} = a^{N} and (x^{m})^{t−τ} = a^{ℓ} . Explicitly, a is given as a = x^{(t−τ)m/ℓ}, from which we conclude that a^{ε+1} = 1. In addition, (x^{m})^{ε+1} = 1 . Changing notation, we write again x instead of x^{m}. Thus we have produced a Kummer radical x of A′|R′ with the properties:

x^{n} = a^{N}  and  x^{t−τ} = a^{ℓ}  and  x^{ε+1} = 1 .

In addition we have a^{ε+1} = 1 which shows that the pair (a, 1) is admissible, as required.

Case 2: ε ∈ 𝔥.

In this case we have to invoke our hypothesis that the decomposition group 𝔥 is cyclic. Hence, since ε is not a square in 𝔥 we conclude that 𝔥 = ⟨ε⟩ is of order 2.

The powers 1, τ, τ^{2}, …, τ^{s−1} represent the cosets of 𝔤 modulo 𝔥 = ⟨ε⟩. Let e be a primitive idempotent of R′. The stabilizer of e in 𝔤 is the decomposition group 𝔥, and the conjugates e, e^{τ} , e^{τ^{2}}, …, e^{τ^{s−1}} are precisely the different primitive idempotents of R′. The decomposition

1 = e + e^{τ} + ⋯ + e^{τ^{s−1}}  (28)

into orthogonal primitive idempotents leads to a direct decomposition

R′ = eR′ × e^{τ} R′ × ⋯ × e^{τ^{s−1}}R′
= eR′ × (eR′)^{τ} × ⋯ × (eR′)^{τ^{s−1}}
≈ S′ × S′ × ⋯ × S′  with S′ := eR′  (29)

This decomposition is the analogue, in Case 2, of the decomposition (26) which we have used in Case 1.

Now let A|R be a Galois G-algebra and A′ = A ⊗_{R} R′. Then 𝔤 acts on A′ such that A = Fix(𝔤, A′). We claim:

There exists a Kummer radical x of A′|R′ such that x = y^{N} with y ∈ A^{′×}. This can be seen as follows: The primitive idempotent e ∈ R′ may not be primitive in A′. Nevertheless, the decomposition (28) yields a decomposition of A′ similarly to (29), namely:

A′ = eA′ × e^{τ} A′ × ⋯ × e^{τ^{s−1}}A′
= eA′ × (eA′)^{τ} × ⋯ × (eA′)^{τ^{s−1}}
≈ B′ × B′ × ⋯ × B′  with B′ := eA′ .  (30)

According to this decomposition, every element x ∈ A′ can be written as a vector

x = (x_{0}, x_{1}, …, x_{s−1})  with  x_{i} ∈ B′ .  (31)

In this representation the automorphism τ acts as the right shift:

x^{τ} = (x_{s−1}, x_{0}, x_{1}, …) .

The automorphism ε acts component-wise:

x^{ε} = (x_{0}^{ε}, …, x_{s−1}^{ε}) .

Note that B′ = eA′ is stable under ε since e is.

We start with an arbitray Kummer radical x of A′|R′. We refer to Statement (iii) of the proof of Prop. 8 for the fact that the element u := x^{t−τ} is contained in R^{′×} (see footnote 10). We write x in the form (31) and similarly u; we conclude that x_{i}^{t}x_{i−1}^{−1} = u_{i} ∈ S^{′×}. By induction it follows x_{s−1}^{t^{s−1−i}} = x_{i}v_{i} (0 ≤ i ≤ s − 1) where the elements v_{i} are power products of the u_{i}; for our purpose it is sufficient to know that v_{i} ∈ S^{′×} and hence v = (v_{0}, v_{1}, …, v_{s−1}) ∈ R^{′×}. Putting y := x_{s−1}, we conclude

(y^{t^{s−1}}, y^{t^{s−2}}, …, y^{t}, y) = (x_{0}, x_{1}, …, x_{s−1})(v_{0}, v_{1}, …, v_{s−1}) = x · v .  (32)

The vector on the left hand side can be written as y^{N} if we identify B^{′×} with the first factor of the product B^{′×} × ⋯ × B^{′×}, so that y is identified with the vector (y, 1, 1, …, 1). Indeed, the definition of the operator N reads N = t^{s−1} + t^{s−2}τ + ⋯ + τ^{s−1} and if we apply this operator to (y, 1, 1, …, 1) we obtain the vector on the left hand side in (32) since τ acts as the shift. So we have

y^{N} = x · v .

Since x is a Kummer radical of A′|R′ and v ∈ R^{′×}, it follows that xv is a Kummer radical too. Changing notation, we write again x instead of xv and thus have obtained a Kummer radical x such that x = y^{N}.

Our above claim is proved. But we have obtained more information. Namely, since y = x_{s−1} appears as the last component of x, it follows that y^{n} is the last component of x^{n}. Since x^{n} ∈ R^{′×} it follows y^{n} ∈ S^{′×}. Let us put a := y^{n} ∈ S^{′×} ⊂ R^{′×}.

Moreover, we know from (27) that x^{ε+1} is contained in R^{′×}. ^{13} Now, y^{ε+1} is the last component of x^{ε+1} and therefore it follows that y^{ε+1} ∈ S^{′×} ⊂ R^{′×}. Let us put c := y^{ε+1}.

We have proved:

There exists a Kummer radical x of A′|R′ such that x = y^{N} for some y ∈ A^{′×}. Moreover, y^{n} = a and y^{ε+1} = c with a, c ∈ R^{′×}.

For this Kummer radical we compute:

x^{n} = y^{nN} = a^{N}
x^{t−τ} = y^{(t−τ)N} = y^{ℓn} = a^{ℓ}
x^{ε+1} = y^{(ε+1)N} = c^{N} ,

these are the relations (25), and

a^{ε+1} = y^{(ε+1)n} = c^{n}
c^{ε−1} = y^{(ε+1)(ε−1)} = 1 .

this shows that the pair (a, c) is admissible.

Proposition 12 is proved.

Again we shall have occasion to consider base extension. Suppose that R is contained in the semi-simple commutative K-algebra S. We assume that the unit element of R is also the unit element of S, hence R^{×} ⊂ S^{×}. Moreover, R′ = K′⊗_{K}R is contained in S′ = K′ ⊗_{K} S and R^{′×} ⊂ S^{′×}. In this situation we have in the exceptional case too:

Lemma 13 (Base extension) Suppose that 𝔤 is exceptional, and that the decomposition groups of the Galois 𝔤-algebra R′|R are cyclic. Let A be a Galois G-algebra over R with Kummer parameter pair (a, c). If a and c are considered as elements in S^{′×}, then (a, c) is a Kummer parameter pair for the Galois G-algebra A ⊗_{R} S over S, which is obtained from A by base extension R ⊂ S.

The proof proceeds as in the non-exceptional case.

The following Lemma gives a method to generate admissible pairs. As introduced above, R′_{ε} = Fix(ε, R′) denotes the subalgebra of elements which are fixed under the exceptional automorphism ε.

Lemma 14 If (a, c) ∈ R^{′×} ×R_{ε}^{′×} is admissible then a can be written in the form a = a^{ε−1}c^{n/2} with ã ∈ R^{′×}. Conversely, if ã ∈ R^{′×} and c ∈ R_{ε}^{′×} are arbitrary, then by putting a = a^{ε−1}c^{n/2}, the pair pair (a, c) is admissible.

In other words: The map (ã, c) ↦ (a, c) as described above is a surjective homomorphism from R^{′×} × R_{ε}^{′×} onto the group W of admissible pairs.

Proof : (a, c) to be admissible means that a^{ε+1} = c^{n} and c^{ε} = c. From this it follows (ac^{−n/2})^{ε+1} = 1. We apply “Hilbert’s Theorem 90” (see A.14) to the Galois ⟨ε⟩-algebra R′|R′_{ε}. We conclude that there exists ã ∈ R^{′×} such that ac^{−n/2} = a^{ε−1}, hence a = a^{ε−1}c^{n/2}.

13In (27) we have only used that ε commutes with every automorphism of G; this is true in both cases, Case 1 and Case 2.

The converse is directly verified.

3.2 Proof of Grunwald-Wang theorem. We consider the situation of Theorem 1 in the exceptional case; this implies in particular that n = 2^{ν} is a power of 2.

Again, K′ = K( \sqrt[n]{1}) denotes the field of n-th roots of unity over K. Let K′_{ε} = Fix(ε, K′) denote the fixed field of the exceptional automorphism ε. Let Let K̂, K̂′_{ε}, K̂′ be the completions of K, K′_{ε}, K′ respectively.

The decomposition groups in K′ of the valuations v ∈ V are precisely the decomposition groups of the Galois 𝔤-algebra K̂′ over K̂. We assume that all those decomposition groups are cyclic. According to Proposition 12 this implies that every Galois G-algebra A over K̂ admits a Kummer parameter pair (a, c).

Similarly as in the non-exceptional case we try to choose a Kummer parameter pair (a, c) for A such that a, c ∈ K^{′×}.

We start with an arbitrary parameter pair (a, c) for A and write a = a^{ε−1}c^{n/2} as in Lemma 14. Since K′ is dense in K̂′ there exists b̃ ∈ K′ which is close to ã. Applying Lemma 10 we conclude that

ã ≡ b̃ mod K̂^{′×n}.

Similarly, since K′_{ε} is dense in K̂′_{ε}, we find d ∈ K′_{ε} close to c, and conclude

c ≡ d mod K̂_{ε}^{′×n}.

Putting b = b̃^{ε−1}d^{n/2} we obtain an admissable pair (b, d) (by Lemma 14) and we have b ∈ K′ and d ∈ K′_{ε}. Moreover, Lemma 14 shows that we have

(a, c) ≡ (b, d) mod W^{n}.

Consequently, Proposition 12(ii) says that the admissible pair (b, d) parametrizes the same algebra A as (a, c).

We change notation and write again (a, c) instead of (b, d). We have shown: There exists a Kummer parameter pair (a, c) for A|K̂ such that a ∈ K^{′×} and

c ∈ K_{ε}^{′×}.

Now we use Proposition 12(i), but over K instead of K̂. Hence (a, c) is a pair of Kummer parameters of a certain Galois G-algebra L over K. We apply Lemma 13 with respect to the base extension K ⊂ K̂ and conclude that L ⊗_{K} K̂ ≈ A. In view of (3) we see that L̂ ≈ A.

4 The case of characteristic p

As before, G is a cyclic group of prime power order n = p^{ν}, and K is a field. Now we consider the case that char(K) = p.

4.1 Preliminaries on Witt vectors. Let A be a commutative ring with unit element such that pA = 0; thus A is an 𝔽_{p}-algebra. We denote by W_{ν}(A) the ring of Witt vectors x = (x_{0}, …, x_{ν−1}) of length ν over A. ^{14} Addition and multiplication of those vectors are defined by polynomials. More precisely, if the Witt vector z is the sum (or product) of x and y then the i-th component z_{i} is a certain polynomial, with integer coefficients, in x_{0}, … x_{i}, y_{0}, …, y_{i}. ^{15}

14For the basic facts about Witt vectors we refer to [6] II. § 26.

The Frobenius operator F on W_{ν}(A) is defined by

F (x) = (x_{0}^{p}, x_{1}^{p}, …) ;

this is an endomorphism of the ring W_{ν}(A). The Artin-Schreier map ℘ on W_{ν}(A) is defined by

℘(x) = F (x) − x .

This map is additive. For ν ≥ 1 the shift operator V : W_{ν−1}(A) → W_{ν}(A) is defined by

V (x_{0}, x_{1}, …, x_{ν−2}) = (0, x_{0}, x_{1}, …, x_{ν−2}) .

This map is useful for induction arguments. ^{16} V is additive and injective, and it satisfies F V = V F. It follows ℘V = V ℘. The image of V is the kernel of the canonical homomorphism W_{ν}(A) → A which maps every vector x onto its first component x_{0}. Thus we have the exact sequence

0 → W_{ν−1}(A) ⟶^{V} W_{ν}(A) ⟶ A → 0

For x_{0} ∈ A we denote by {x_{0}} the vector ^{17}

{x_{0}} = (x_{0}, 0, 0, …, 0) .

The map x_{0} ↦ {x_{0}} is a section for the projection homomorphism W_{ν}(A) → A; it is multiplicative but not additive. The fundamental rules for the addition of Witt vectors imply ^{18} for x = (x_{0}, x_{1}, …, x_{ν−1}) that

x = {x_{0}} + V x̃ with  x̃ = (x_{1}, x_{2}, …, x_{ν−1}) ∈ W_{ν−1}(A) .  (33)

We shall have to use this formula in the sequel.

4.2 Witt radicals for cyclic Galois algebras. Now let R be a semisimple commutative K-algebra. We want to give a description of the Galois G-algebras over R.

We consider W_{ν}(𝔽_{p}) as a subring of W_{ν}(R). The additive group W_{ν}(𝔽_{p})^{+} is well known to be cyclic of order p^{ν} (see [6] II. § 26 p.147). We choose an isomorphism

χ: G → W_{ν}(𝔽_{p})^{+} .

This isomorphism is kept fixed throughout and all statements in the sequel refer to the given χ.

Let a = (a_{0}, …, a_{ν−1}) ∈ W_{ν}(R). Consider the R-algebra A = R[x] generated by ν elements x_{0}, …, x_{ν−1} which, when interpreted as a vector x = (x_{0}, …, x_{ν−1}) ∈ W_{ν}(A), satisfy

℘(x) = a  (34)

as their defining relations. This means that A is isomorphic to the factor algebra of the polynomial algebra R[X] = R[X_{0}, …, X_{ν−1}] modulo the ideal I generated by the polynomials f_{0}, …, f_{ν−1} ∈ R[X] which describe the relation (34), i.e.,

℘(X) − a = (f_{0}(X), …, f_{ν−1}(X))  in  W_{ν}(R[X]) ;

15For later use we note that these polynomials have vanishing constant coefficients.

16In our notation of F,V and ℘ we suppress the dependence on ν; even for different ν we shall use the same symbols F,V and ℘.

17This is Witt’s original notation [14].

18See [6] II. § 26 p.140

and x_{i} denotes the image of X_{i} in A (0 ≤ i ≤ ν − 1). In this sense, the relation (34) in W_{ν}(A) is to be interpreted as a system of polynomial relations

f_{i}(x_{0}, …, x_{ν−1}) = 0 (0 ≤ i ≤ ν − 1)

and these are defining relations for the generators x_{0}, …, x_{ν−1} of A|R.

If we assign to the generating vector x = (x_{0}, …, x_{ν−1}) some vector y = (y_{0}, …, y_{ν−1}) with components y_{i} ∈ A satisfying the same relation ℘(y) = a, then this defines uniquely an R-algebra homomorphism of A into itself which maps the x_{i} onto the y_{i}. ^{19} Let σ ∈ G. We have ℘(χ(σ)) = 0 since χ(σ) ∈ W_{ν}(𝔽_{p}). Hence for y = x + χ(σ) we have ℘(y) = ℘(x) = a. Thus σ defines an R-algebra homomorphism of A into itself. By general functorial properties of Witt vectors, this extends canonically to a ring homomorphism of W_{ν}(A) into itself, denoted also by σ, and we have

x^{σ} = (x_{0}^{σ}, …, x_{ν−1}^{σ}) .  (35)

So the definition of the action of σ on A can be put into the formula

x^{σ} = x + χ(σ)  (σ ∈ G)  (36)

(which again is to be interpreted as a system of polynomial equations for the components of the respective vectors). The homomorphism property χ(στ) = χ(σ)+χ(τ) yields x^{στ} = (x^{σ})^{τ}. In this way we see that G acts on A and also on W_{ν}(A).

Thus the relations (36) define on A the structure of G-algebra over R. We denote this algebra by A_{a} since it is uniquely determined by the Witt vector a ∈ W_{ν}(R).

The following result, valid in characteristic p, is Witt’s additive analogue to the multiplicative Kummer Theory of Proposition 4.

Proposition 15 (Witt Theory for Galois algebras) (i) Let a ∈ W_{ν}(R). Then any G-algebra A|R generated by the components of some Witt vector x ∈ W_{ν}(A) satisfying the relations

℘(x) = a  and  x^{σ} = x + χ(σ) ,  (σ ∈ G)  (37)

is a Galois G-algebra, and A is isomorphic to A_{a} (as G-algebras over R).

(ii) If a ≡ b mod ℘W_{ν}(R), i.e., if a = b+℘(u) with u ∈ W_{ν}(R) then the Galois G-algebra A_{a} determined by a is isomorphic to the Galois G-algebra A_{b} determined by b. And conversely.

(iii) Every Galois G-algebra A|R is of the type described in (i), i.e., A is isomorphic to A_{a} (as Galois G-algebra over R) for suitable a ∈ W_{ν}(R). Such a Witt vector a is called a Witt radicand of A, and the corresponding x is a Witt radical of A.

Remark: We can reformulate Proposition 15 as follows:

Every Witt vector a ∈ W_{ν}(R) defines (uniquely up to isomorphisms) a Galois G-algebra A|R such that a is a Witt radicand of A. The structure of A depends only on the (additive) residue class of a modulo ℘W_{ν}(R). Conversely, every Galois G-algebra A|R admits a vector a ∈ W_{ν}(R) as its Witt radicand.

Proof of (i): We shall use induction with respect to ν.

19In other words: the map R[X] → A given by X_{i} ↦ y_{i} (0 ≤ i ≤ ν − 1) can be factored through R[X]/I ≈ R[x].

Using A.3 (and basic functorial properties of Witt vectors) we may suppose that R is a field. To show that A is a Galois G-algebra we use A.4. Hence we have to show that

(a) A is semisimple.

(b) Fix(G, A) = R.

(d) G acts faithfully on A. Statement (d) is immediate from the second relation in (37). It remains to verify (a) and (b).

First consider the case ν = 1. Then W_{1}(A) can be identified with A itself; in the relation (37) we now have x ∈ A and a ∈ R. Thus the generator x of A|R satisfies an Artin-Schreier equation in the usual sense: x^{p} − x = a. Let f(X) = X^{p} − X − a ∈ R[X] (one variable X). Then we have A_{a} ≈ R[X]/f(X). Since f(X) is separable (i.e., it has no multiple roots), it follows that A_{a} is semisimple; this gives (a) for A_{a}. If f(X) is irreducible over R then A_{a}|R is a Galois extension of fields of degree p, and assertion (b) follows for A_{a}. If f(X) is not irreducible over R then it splits completely in R[X]:

f(X) = (x − ϑ_{1}) ⋯ (x − ϑ_{p})

where ϑ_{1}, …, ϑ_{p} ∈ R are the roots of f(X). It follows that

A_{a} ≈ R × ⋯ × R

this decomposition is defined by assigning to x the vector (ϑ_{1}, …, ϑ_{p}) and, accordingly, to every polynomial h(x) ∈ R[x] = A_{a} the vector (h(ϑ_{1}), …, h(ϑ_{p})). Here, h(x) may be assumed to be of degree < p. The automorphisms σ ∈ G permute the ϑ_{i} cyclically. If h(x) is fixed under all σ ∈ G then h(ϑ_{i}) = h(ϑ_{1}) for all i. It follows that h(x) is a constant polynomial, i.e., h(x) ∈ R. Thus we have (b) for A_{a}.

We have now shown (for ν = 1) that A_{a} is a Galois G-algebra over R. As to A, since it is generated by x over R, there is a natural surjective G-homomorphism A_{a} → A as G-algebras; applying A.12 it follows that this is an isomorphism and, hence, A too is a Galois G-algebra.

Now suppose ν > 1. Consider the projection homomorphism W_{ν}(A) → A given by (x_{0}, …, x_{ν−1}) ↦ x_{0}. Then A = R[x] is projected onto A_{0} = R[x_{0}] with the relations

℘(x_{0}) = a_{0} and x_{0}^{σ} = x_{0} + χ_{0}(σ) (σ ∈ G)

where χ_{0}(σ) ∈ 𝔽_{p} denotes the first component of the Witt vector

χ(σ) = (χ_{0}(σ), …, χ_{ν−1}(σ)) .

We have χ_{0}(σ^{i}) = i · χ_{0}(σ) = 0 if and only if i ≡ 0 mod p. Hence the group G^{p} of p-th powers in G acts trivially on A_{0} while the factor group G/G^{p} acts faithfully on A_{0}. By what we have seen in the case ν = 1, A_{0} is a Galois G/G^{p}-algebra over R. In particular, A_{0} is semisimple and we have

Fix(G/G^{p}, A_{0}) = R .  (38)

Now consider A as a G^{p}-algebra over A_{0}. We claim that A|A_{0} is a Galois G^{p}-algebra.

A is generated over A_{0} = R[x_{0}] by x_{1}, …, x_{ν−1} which we regard as the components of the Witt vector x̃ = (x_{1}, …, x_{ν−1}) ∈ W_{ν−1}(A), of length ν − 1. Let us put b := ℘(x̃). Using (33) we compute

V b = V ℘(x̃) = ℘V (x̃) = ℘(x − {x_{0}}) = a − ℘{x_{0}}

from which we infer that the components of V b, hence those of b, are polynomials (with integer coefficients) in a_{0}, a_{1}, …, a_{ν−1} and x_{0}. We conclude that the components of b are contained in A_{0} = R[x_{0}]. Thus we have

℘(x̃) = b ∈ W_{ν−1}(A_{0}).

A similar computation for the action of an automorphism τ ∈ G^{p} leads to the following: We have already seen above that the first component χ_{0}(τ) = 0 since τ = σ^{p} ∈ G^{p}. Thus

χ(τ) = (0, χ_{1}(τ), …, χ_{ν−1}(τ)) = V χ̃(τ)

where χ̃(τ) = (χ_{1}(τ), …, χ_{ν−1}(τ)) ∈ W_{ν−1}(𝔽_{p}) . Now we observe that the operator τ acts componentwise, i.e., x^{τ} = (x_{0}^{τ}, …, x_{ν−1}^{τ}). We have seen above already that τ ∈ G^{p} acts trivially on A_{0}, hence x_{0}^{τ} = x_{0}. We compute, using (33):

V x^{τ} = x^{τ} − {x_{0}^{τ}} = x + χ(τ) − {x_{0}}
= (x − {x_{0}}) + χ(τ)
= V x̃ + V χ̃(τ) = V (x̃ + χ̃(τ))

and hence

x^{τ} = x̃ + χ̃(τ) .

We have seen that A, as a G^{p}-algebra over A_{0}, is generated by the components of the Witt vector x̃ of length ν − 1, satisfying relations of the same form as does x over R, namely:

℘(x̃) = b ∈ W_{ν−1}(A_{0})  and  x^{τ} = x̃ + χ̃(τ) , (τ ∈ G^{p}) .

By induction hypothesis we conclude that, indeed, A is a Galois G^{p}-algebra over A_{0}.

In particular A is semisimple, which gives (a). Moreover, A_{0} = Fix(G^{p}, A) and hence, using (38):

Fix(G, A) = Fix(G, A_{0}) = Fix(G/G^{p}, A_{0}) = R .

This gives (b).

We have now shown that A is a Galois G-algebra over R. It follows that A_{a} too is a Galois G-algebra over R, since A_{a} satisfies the same hypotheses as announced in the statement of the proposition. Now again, since A is generated by x there is a natural surjection A_{a} → A, and we infer from A.12 that this is an isomorphism.

Proof of (ii): Consider the algebra A_{a} = R[a] with (37), and similarly A_{b} = R[y]. Suppose that a = b + ℘(u) with u ∈ W_{ν}(R). We are going to establish an isomorphism φ : A_{a} → A_{b} as G-algebras over R.

The relation ℘(x) = a is a defining relation for x over R. Hence, in order to obtain a homomorphism φ : A_{a} → A_{b} as R-algebras, it is sufficient to assign to x some vector z ∈ W_{ν}(A) such that z satisfies the same relation ℘(z) = a as x does. Clearly this holds for z := y + u since ℘(y + u) = b + ℘(u) = a.

Hence we have a uniquely defined R-algebra homomorphism φ : A_{a} → A_{b} such that x^{φ} = y + u. We claim that this is a homomorphism as G-algebras. Indeed: for σ ∈ G we have x^{σφ} = (x + χ(σ))^{φ} = x^{φ} + χ(σ) = y + u + χ(σ) = y^{σ} + u = (y + u)^{σ} = x^{φσ} since σ acts trivially on u ∈ W_{ν}(R).

Since both A_{a} and A_{b} are Galois G-algebras over R it follows that φ : A_{a} → A_{b} is an isomorphism. (See A.12.)

Conversely, assume that A_{a} and A_{b} are isomorphic Galois G-algebras. Let us identify A_{a} = A_{b} = A by that isomorphism. Thus on the one hand, A_{a} = R[x] with (37), and on the other hand A_{a} = A_{b} = R[y] with corresponding relations for y and b. When applying σ ∈ G both x and y take the same additive summand, namely χ(σ). We conclude (y − x)^{σ} = y − x for all σ ∈ G. It follows that the coordinates of the vector u := y − x are contained in R, i.e., u ∈ W_{ν}(R). We have b = ℘(y) = ℘(x) + ℘(u) = a + ℘(u) ≡ a mod ℘R .

Proof of (iii): Now let A be an arbitrary Galois G-algebra over R. We have to find x ∈ W_{ν}(A) and a ∈ W_{ν}(R) satisfying the relations (37), and such that A is generated over R by the components of x.

Since G acts on A it also acts on the ring W_{ν}(A) of Witt vectors by means of (35). Consider the trace operator S on W_{ν}(A) defined by

S(u) = ∑_{τ∈G} u^{τ}  for  u ∈ W_{ν}(A) .

Since the vector S(u) is fixed by every σ ∈ G, its components are fixed by G and hence are contained in R; we conclude that S(u) ∈ W_{ν}(R). We claim: There exists u ∈ W_{ν}(A) such that S(u) is a unit in W_{ν}(R).

To see this we recall that a vector is a unit in W_{ν}(R) if and only if its first component is a unit in R.^{20} Now the first component of S(u) is S(u_{0}) where u_{0} ∈ A is the first component of u. Since A is a Galois G-algebra it admits a normal basis over R. If we choose u_{0} ∈ A as a generator of such a normal basis then S(u_{0}) ∈ R^{×}. For, if this were not the case then there would exist c ≠ 0 in R such that

c · S(u_{0}) = ∑_{τ} c · u_{0}^{τ} = 0

which contradicts the fact that the u_{0}^{τ} form a basis of A over R.

Thus indeed, there exists a vector u ∈ W_{ν}(A) such that S(u) ∈ W_{ν}(R)^{×}; we have to choose u such that its first component u_{0} is a generator of a normal basis of A|R. After replacing u by S(u)^{−1}u we may assume that

S(u) = 1 .

We start from such u and put

x := ∑_{τ∈G} χ(τ^{−1})u^{τ} ,

like we did in the multiplicative Kummer theory. We compute for σ ∈ G:

x^{σ} = ∑_{τ∈G} χ(τ^{−1})u^{τσ} = ∑_{τ∈G} χ(τ^{−1}σ)u^{τ}

= ∑_{τ∈G}(χ(τ^{−1}) + χ(σ))u^{τ}

= ∑_{τ∈G} χ(τ^{−1})u^{τ} + χ(σ) ∑_{τ∈G} u^{τ}

= x + χ(σ) .

Let us put a := ℘(x). Then we compute

a^{σ} = ℘(x^{σ}) = ℘(x + χ(σ)) = ℘(x) = a

20See e.g., Witt [14], p.131, Satz 5.

since ℘(χ(σ)) = 0. Hence the components of a are fixed under G and therefore contained in R; it follows a ∈ W_{ν}(R).

Thus the vector x ∈ W_{ν}(A) satisfies relations of the form (37). From (i) we conclude that the subalgebra R[x] ⊂ A is a Galois G-algebra, with the action of G on R[x] induced by the action of G on A. The inclusion map R[x] ↪ A is a map of Galois G-algebras and hence an isomorphism by A.12; this shows that R[x] = A.

Remark: Let A|R be a Galois G-algebra and a ∈ W_{ν}(R) a Witt radicand of A. The corresponding Witt radical x ∈ W_{ν}(A) of a is not uniquely determined. An element y ∈ W_{ν}(A) is another Witt radical of a if and only if y = x + c with c ∈ W_{ν}(R) and ℘(c) = 0. If R is a direct product of d fields then there are n^{d} such elements c. The n^{d} substitutions x ↦ x + c yield n^{d} automorphisms of A as Galois G-algebra over R, and every automorphism of A is of this form.

Similarly as in multiplicative Kummer theory, we shall need the following lemma concerning base extension. Suppose that R is contained in the semi-simple commutative K-algebra S. We assume that the unit element of R is also the unit element of S. Then W_{ν}(R) ⊂ W_{ν}(S). In this situation we have:

Lemma 16 (Base extension) Let A be a Galois G-algebra over R with Witt radicand a ∈ W_{ν}(R). If a is considered as a vector in W_{ν}(S), then it is a Kummer parameter for the Galois G-algebra A ⊗_{R} S over S, which is obtained from A by base extension R ⊂ S. ^{21}

The proof is again immediate: We have A ⊂ A ⊗_{R} S and hence W_{ν}(A) ⊂ W_{ν}(A ⊗_{R} S). If x ∈ W_{ν}(A) satisfies the relations (37) then it satisfies the same relations when considered as a vector in W_{ν}(A ⊗_{R} S).

4.3 Proof of Grunwald-Wang theorem. We consider the situation of the Grunwald-Wang theorem; thus K is a multi-valued field and K̂ its completion. We assume that char(K) = p. We have to use the following Lemma, which is the additive analogue to the corresponding Lemma 10 in the multiplicative case.

Lemma 17 Suppose that n = p^{ν} where p is the characteristic of K. If the components z_{i} ∈ K̂ of the Witt vector z ∈ W_{ν}(K̂) are sufficiently close to 0 then there exists u ∈ W_{ν}(K̂) such that z = ℘(u).

Proof : K̂ is a direct product of the complete fields K̂_{v}. It suffices to discuss each factor K̂_{v} separately. In other words: we may assume K̂ to be a complete field with a single valuation |·|. Note that the valuation is non-archimedean since char(K) = p.

We claim that the condition

|z_{i}| < 1  for  0 ≤ i ≤ ν − 1

is already sufficient: if this condition is satisfied then we claim there exists a vector u = (u_{0}, …, u_{ν−1}) ∈ W_{ν}(K̂) with ℘(u) = z. Moreover, u can be chosen such that |u_{i}| < 1 for 0 ≤ i ≤ ν − 1. We use induction on ν.

If ν = 1 we have W_{1}(K̂) = K̂. We use Hensel’s lemma for the polynomial f(X) = X^{p} − X − z ∈ K̂[X]. If u is a zero of f(X) then z = ℘(u). We have

|f(0)| = |z|  and  |f′(0)| = 1.

21See A.10 for the fact that A ⊗_{R} S is a Galois G-algebra over S.

Therefore, if |z| < 1 then Hensel’s lemma guarantees the existence of a unique u ∈ K̂ such that f(u) = 0 and |u| < 1.

Now suppose ν > 1 and write (see (33)):

z = {z_{0}} + V z̃.

Using what we have just seen for ν = 1, we let u_{0} ∈ K̂ with |u_{0}| < 1 such that ℘(u_{0}) = z_{0}. Using the induction hypothesis we find ũ ∈ W_{ν−1}(K̂) such that ℘(ũ) = z̃, and each component of ũ has value < 1. Now we write

z = {℘ u_{0}} + V ℘(ũ)
= ℘ ({u_{0}} + V ũ) + ({℘ u_{0}} − ℘{u_{0}})
= ℘u + t

if we put

u := {u_{0}} + V ũ and t := {℘ u_{0}} − ℘{u_{0}} .

Thus the vector u as constructed does not yet solve our requirements; we still have to discuss the remainder t.

Let us write t = {t_{0}} + V t̃. By definition of t we have t_{0} = ℘(u_{0}) − ℘(u_{0}) = 0 and therefore t = V t̃with t̃= (t_{1}, …, t_{ν−1}) ∈ W_{ν−1}(K̂). We can use the induction hypothesis and conclude that t̃ = ℘(u′) with u′ ∈ W_{ν−1}(K̂) – provided we know that |t_{i}| < 1 for all i. Now, from the definition of t we see that each t_{i} is a polynomial in u_{0} with integer coefficients. Since these polynomials do not have constant coefficients ^{22} and since |u_{0}| < 1 we conclude that each |t_{i}| < 1. Recall that the valuation of K̂ is non-archimedean.

Thus by the induction hypothesis we have t̃ = ℘(u′) with u′ ∈ W_{ν−1}(K̂), and all components of u′ have value |u′_{i}| < 1. It follows t = V ℘(u′) = ℘(V u′) and hence

z = ℘(u) + ℘(V u′) = ℘(u + V u′) .

This proves our assertion since, by footnote 15 again, all components of u + V u′ have value < 1.

Proof of Grunwald-Wang theorem:

Besides of K we consider its completion K̂. We apply Prop.15 to Galois Galgebras over R = K̂.

Let A be a Galois G-algebra over K̂, and let a ∈ W_{ν}(K̂) be a Witt radicand for A, according to Prop.15(iii). We observe that K is dense in K̂; hence there are elements b_{i} ∈ K which are arbitrarily close to the components a_{i} of a (0 ≤ i ≤ ν−1). Then the a_{i} − b_{i} are close to 0 and we infer from Lemma 17 that b − a ≡ 0 mod ℘W_{ν}(K̂). Hence by Prop.15(ii), b = (b_{0}, …, b_{ν−1}) is also a Witt radicand for A. Changing notation, we have seen:

There exists a Witt radicand a for A|K̂ which is contained in W_{ν}(K).

Now, we use Prop.15(i), but over K instead of K̂. Hence a is a Witt radicand of a certain Galois G-algebra L over K. We apply Lemma 16 with respect to the base extension K ⊂ K̂ and conclude that L ⊗_{K} K̂ ≈ A. In view of (3) we see that L̂ ≈ A.

22See footnote 15.

5 Appendix: On Galois algebras

All algebras considered here are supposed to be algebras over a field, and to have a unit element. ^{23}

Let R be a semisimple commutative algebra and G a finite group. A G-algebra over R is defined to be a commutative R-algebra A, together with an action of G on A, such that every σ ∈ G acts as an R-algebra automorphism of A. We use the exponential notation for this action, thus x^{σ} denotes the image of x ∈ A under the action of σ ∈ G. Every such G-algebra carries the structure of a right RG-module, where RG denotes the group ring of G over R.

A.1 Definition: A G-algebra A|R is called Galois G-algebra if the

following two conditions are satisfied:

(a) A is semisimple.

(b) A is a free RG-module of rank 1.

Property (b) is equivalent to the existence of a normal basis for A|R, i.e., the existence of an element u ∈ A such that its G-images u^{σ} (σ ∈ G) form an R-basis of A. In particular it follows that G acts faithfully on A. Moreover, the existence of an R-basis of A implies that we can identify R = R · 1_{A} with a subalgebra of A. Then R = Fix(G, A), the fixed algebra of G in A, consisting of those x ∈ A which are fixed under all σ ∈ G.

A.2 Note: Every Galois extension of fields L|K with Galois group G is a Galois G-algebra in a natural way. (Normal Basis theorem, cf. [6] §12.)

Any commutative semisimple algebra R is the direct product of fields:

R = K_{1} × ⋯ × K_{m}

where the component fields K_{i} are uniquely determined as subsets of R (cf. [6], §29). If e_{i} denotes the unit element of K_{i}, then

1 = e_{1} + ⋯ + e_{m}  with  e_{i}e_{j} = \begin{cases}e_{i} & if i = j \\ 0 & if i ≠ j \end{cases}

We have K_{i} = e_{i}R, and as an R-module we have the direct sum decomposition

R = e_{1}R ⊕ ⋯ ⊕ e_{m}R = K_{1} ⊕ ⋯ ⊕ K_{m} .

Every idempotent e ∈ R is a partial sum of e_{1}, …, e_{m}. An idempotent of R is called primitive if it cannot be written as a sum of two orthogonal idempotents of R. The e_{i} are precisely the primitive idempotents of R. In the following we use the notation

P (R) := {e_{1}, …, e_{m}}

for the set of all primitive idempotents in R.

A.3 Reduction Lemma: Let A|R be a G-algebra. Consider the direct product decomposition

A = ∏_{e∈P (R)} eA .  (A1)

For A|R to be a Galois G-algebra it is necessary and sufficient that eA|eR is a Galois G-algebra for each e ∈ P (R).

23For the general terminology and the basic facts which are used without citation we refer to [6], in particular §§28–29.

For the proof one observes that each of the definining conditions (a) and (b) is satisfied for A|R if and only if it is satisfied for all components eA|eR in (A1).

Note that for the algebra eA|eR the ground ring eR is a field. Hence, in the discussion of Galois G-algebras A|R, Lemma A.3 will often allow us to reduce the discussion to the case when R is a field; this explains the name “Reduction Lemma.”

Remark: In the situation of A.3 each eA is also an R-subalgebra of A. However, if R is not a field (i.e., if e ≠ 1) then eA|R is not a Galois G-algebra.

A.4 Galois algebras over a field: Suppose R = K is a field. Let A|K be a G-algebra satisfying the following three conditions:

(a) A is semisimple

(b) Fix(G, A) = K

(c) [A : K] ≥ |G|

where [A : K] denotes the K-dimension of A, and |G| the order of G. Then A|K is a Galois G-algebra and, in fact, we have [A : K] = |G|. If G is abelian then the condition (c) can be replaced by the following condition:

(d) G acts faithfully on A.

We shall prove this together with the following structure theorem for Galois algebras. In fact, we shall give the proof of the structure theorem under the hypotheses of A.4, and then conclude that A is indeed a Galois G-algebra.

A.5 Structure of Galois algebras over a field. Suppose R = K is a field. Let A|K be a G-algebra satisfying (a), (b), (c) above (or (a), (b), (d) if G is abelian). Then G acts transitively on the set P (A) of primitive idempotents of A. Let e ∈ P (A) and put L := eA; this is a field. Then

A = ∏_{σ∈G mod G_{e}} L^{σ} .  (A2)

Here, G_{e} denotes the subgroup of those elements in G which leave e fixed, and σ ranges over a set of representatives of left cosets G_{e}σ of G.

The group G_{e} operates on L and with this operation, L is a Galois G_{e}-algebra over eK. In other words: If L is regarded as an extension field of K (by identifiying K = eK), then L is Galois over K and the action of G_{e} on L yields an isomorphism G_{e} ≈ Gal(L|K).

Proof : The sum s = ∑_{σ∈G mod G_{e}} e^{σ} is left fixed by G. Using (b) we conclude s ∈ K. Since s is an idempotent and K is a field it follows s = 1. Hence 1 ∈ A is the sum of the distinct primitive idempotents e^{σ}. It follows

P (A) = {e^{σ} : σ ∈ G mod G_{e}}

and we obtain the decomposition (A2). The fact that eA is a field follows from hypothesis (a), that A is semi-simple.

Suppose a ∈ L is left fixed by G_{e}. Then b := ∑_{σ∈G mod G_{e}} a^{σ} is left fixed by G. From (b) it follows b ∈ K and hence a = eb ∈ eK. Thus eK = Fix(G_{e}, L). It follows that L|eK is a Galois extension of fields, and its Galois group is induced by the action of G_{e} on L. Thus the Galois group of L|eK is a homomorphic image of G_{e} and therefore

[L : K] ≤ |G_{e}| .  (A3)

It remains to show that G_{e} acts faithfully on L, which by Galois theory is equivalent to [L : K] = |G_{e}|. If we consider A as a K-module then

A = ⊕ ∑_{σ∈G mod G_{e}} L^{σ} ,  (A4)

the direct sum of the K-modules L^{σ}. Comparing K-dimensions we get from (A3)

[A : K] = (G : G_{e}) · [L : K] ≤ (G : G_{e}) · |G_{e}| = |G| .

From (c) we obtain [L : K] = |G_{e}|.

If G is abelian then, using (d) we have the following argument to show that G_{e} acts faithfully on L: Let τ ∈ G_{e} and suppose that a^{τ} = a for each a ∈ L. Then for any σ ∈ G we have a^{στ} = a^{τσ} = a^{σ}, and hence τ leaves every a^{σ} fixed (a ∈ L, σ ∈ G). From (A4) we conclude that τ leaves every element in A fixed. Hence from (d): τ = 1.

Proof of A.4: We have to verify that A admits a G-normal basis over K. We use the structure theorem A.5 which we have proved under the hypotheses of A.4. The field L as defined by the structure theorem is a Galois field extension of K whose Galois group we may identify with G_{e}. There exists a normal basis of L|K. Accordingly let u ∈ L be such that the u^{τ} with τ ∈ G_{e} form a basis of L|K. Each element in G has a unique representation of the form τσ with τ ∈ G_{e}, and σ ranging over a set of representatives of left cosets of G modulo G_{e}. Looking at (A4) we see that the u^{τσ} form a K-basis of A.

A.6 Decomposition groups. The subgroup G_{e} ⊂ G in A.5 is called the decomposition group of A|K associated to the primitive idempotent e.

The decomposition group associated to e^{σ} is σ^{−1}G_{e}σ. Hence the Galois Galgebra A|K determines a class of conjugate subgroups of G as its decomposition groups. If G is abelian then the decomposition group is uniquely determined by A and is called the decomposition group of the Galois G-algebra A|K; it is denoted by G_{A}.

This definition applies only if the ground ring of A is a field. For a Galois Galgebra A|R over an arbitrary semisimple algebra R, we consider the decomposition (A1). The primitive idempotents of R will now be denoted by ε. Thus A is the direct product of the Galois G-algebras εA|εR, ε ∈ P (R). Each εR is a field, and hence εA|εR determines a decomposition group contained in G, unique up to conjugates. Thus, any Galois G-algebra A|R determines finitely many classes of conjugate decomposition groups in G. Every such class corresponds to a primitive idempotent ε ∈ P (R).

A.7 Induced algebras: Let A|K be a Galois G-algebra over a field K. The formula (A2) shows that as G-algebra over K, A is “induced ” by the G_{e}-algebra L|K ^{24} which in turn is a Galois extension of fields whose Galois group is isomorphic to G_{e}. In particular we see that A as G-algebra over K is uniquely determined by the Galois field extension L|K and the isomorphism Gal(L|K) ≈ G_{e} ⊂ G.

24Here we use the notion of “induced G-algebra” in a similar way as the notion of “induced G-module” is used in representation theory. For the latter, we have to regard A as a G-module over K and the formula (A4) shows that and how A is obtained from the G_{e}-module L over K.

If, besides of the addition in A we consider also the multiplication, i.e., if we consider A not only as G-module over K but also as G-algebra over K, and similarly for L, then we speak of “induced G-algebra”. In this case we prefer to use the notation of direct product instead of direct sum, i.e., we use formula (A2) instead of (A4).

Conversely, if L|K is an arbitrary Galois extension of fields and Gal(L|K) ↪ G is any injection of its Galois group into G then the construction (A2) yields a G-algebra A|K; this is a Galois G-algebra as follows from Proposition A.2. Its decomposition group is, up to conjugates, the image of Gal(L|K) in G. In this sense, the theory of Galois G-algebras over a field K is essentially the same as the theory of Galois extensions L|K, together with an injection Gal(L|K) ↪ G.

A.8 Subalgebras: If A|R is a Galois G-algebra, then every R-subalgebra B ⊂ A is semisimple.

To see this, we observe that B is commutative, has no nilpotent elements and is finitely generated over R (hence Artinian) since A has these properties. ^{25}

A.9 Fixed subalgebras: Let A|R be a G-algebra. For a normal subgroup H ⊂ G let B := Fix(H, A) be the corresponding fixed algebra. B is naturally a G/H-algebra over R, and A an H-algebra over B. We claim: If A|R is a Galois G-algebra then:

(i) B|R is a Galois G/H-algebra, (ii) A|B is a Galois H-algebra.

Proof : (i) Choose a normal basis u^{σ} (σ ∈ G) of A|R . Let ϱ run through a set of representatives for the cosets ϱH of G modulo H. The elements u_{ϱ} := ∑_{τ∈H} u^{ϱτ} are fixed under H and hence contained in B. A direct verification shows that these u_{ϱ} form an R-basis of B. Since H is normal in G, the left and right cosets of G modulo H coincide: ϱH = Hϱ. Hence for the basis constructed above, we have u_{ϱ} = u_{1}^{ϱ}. Thus the u_{ϱ} form a G/H-normal basis of B|R. Since B is semisimple by A.8, it follows that B|R is a Galois G/H-algebra.

(ii) Next we show that A is a Galois H-algebra over B. In view of the reduction lemma A.3 we may assume from the start that R = K is a field.

(iia) If B is a field too then we simply apply A.4 to the H-algebra A over the field B. In our case, condition (c) of A.4 reads [A : B] ≥ |H|. This condition is satisfied because on the one hand |G| = [A : K] = [A : B] · [B : K], and on the other hand we know from (i) that [B : K] = (G : H) (since B admits the K-basis u_{ϱ} corresponding to the cosets ϱH of G modulo H); it follows [A : B] = |H|.

(iib) In general, however, B is not a field and we have to discuss the structure of B in more detail. Again using A.3 it suffices to show that for every primitive idempotent ε ∈ P (B) the H-algebra εA is a Galois H-algebra over εB. Here, εB is a field. We also note that εB = εFix(H, A) = Fix(H, εA). We shall see in (iic) that εA|εK is a Galois G_{ε}-algebra for some subgroup G_{ε} ⊂ G containing H. Thus for εA and the group G_{ε} we have precisely the situation as discussed above in (iia), and we conclude that, indeed, εA is a Galois H-algebra over εB.

(iic) The primitive idempotent ε of B need not be primitive in A. In any case ε is a sum of orthogonal primitive idempotents of A; let e be one of them. Since ε is fixed under H, all H-conjugates e^{τ} with τ ∈ H are also summands of ε. Consider the sum ∑′_{τ∈H} e^{τ} where the prime indicates that each H-conjugate of e appears only once. The above sum is an idempotent fixed under H, hence is contained in B. Since ε is primitive in B it follows

25See [6] §29, Satz 2.

ε = ∑_{τ∈H} ′e^{τ} .  (A5)

To say that each H-conjugate of e appears only once in ε, is equivalent to saying that τ ranges over a set of coset representatives of H modulo G_{e} ∩ H. Here, the notation is the same as in A.5, namely G_{e} = Fix(e, G). These τ form a set of representatives of G_{e}H modulo G_{e}; note that G_{e}H is a group since H is normal in G. We put G_{ε} := G_{e}H. It is straightforward to verify that G_{ε} = Fix(ε, G).

εA is naturally a G_{ε}-algebra, and we have from (A5)

εA = ∏_{τ∈G_{ε} mod G_{e}} e^{τ} A = ∏_{τ∈G_{ε} mod G_{e}} L^{τ}  (A6)

where we have put L := eA; this is a field. We conclude that the G_{ε}-algebra εA is induced from the G_{e}-algebra L|eK, the latter being a Galois extension of fields with Galois group G_{e} according to A.5 (applied to the original Galois G-algebra A over K). We conclude from A.7 that εA is a Galois G_{ε}-algebra over εK, as contended.

A.10 Base extension: Let A|R be a Galois G-algebra. Let S be a

commutative semisimple R-algebra. Then A ⊗_{R} S is a Galois G-algebra over S, the action of G on A ⊗_{R} S being defined by (x ⊗ y)^{σ} = x^{σ} ⊗ y. Moreover:

(i) For any subgroup H ⊂ G we have Fix(H, A ⊗_{R} S) = Fix(H, A) ⊗_{R} S.

(ii) Each of the decomposition groups of A ⊗_{R} S is contained in some decomposition group of A.

Proof : Starting with a normal R-basis u^{σ} (σ ∈ G) for A we obtain a normal S-basis u^{σ} ⊗ 1 for A ⊗_{R} S.

For a subgroup H ⊂ G we use the notation of the proof of A.9; the u_{ϱ} as defined there form an R-basis of Fix(H, A) and the u_{ϱ} ⊗ 1 an S-basis of Fix(H, A) ⊗_{R} S, where ϱ ranges over a set of representatives of the cosets of G modulo H.

To show that A ⊗_{R} S is semisimple, we first decompose S into a direct product of fields; note that S is assumed to be semisimple. This reduces the proof to the case where S, and hence R too, is a field.

Secondly, we decompose A into a direct product of fields: A = ∏_{e∈P (A)} eA . By the structure theorem, each eA is a Galois field extension of eR, and its Galois group is one of the decomposition groups of A|R. This reduces the proof to the case where A|R is a Galois extension of fields, and G the Galois group of this extension.

Now, the tensor product A ⊗_{R} S of two field extensions A|R and S|R, one of which is algebraic, separable and of finite degree, is well known to be a direct product of finitely many fields. Hence A ⊗_{R} S is semisimple. In fact, each direct field component of A ⊗_{R} S is the field compositum A · S after an embedding of A into an algebraically closed overfield of S. It is well known that A · S is a Galois extension of S, its Galois group being naturally isomorphic to the Galois group of A|A ∩ S, hence a subgroup of G.

A.11 Tensor products: (i) If A|R is a Galois G-algebra and B|R a

Galois H-algebra then A ⊗_{R} B is a Galois G × H-algebra. Here, the action of G × H on A ⊗_{R} B is defined by (x ⊗ y)^{(σ,τ)} = x^{σ} ⊗ y^{τ}.

(ii) Conversely, every Galois G × H-algebra C over R is isomorphic to the tensor product A ⊗_{R} B of the Galois G-algebra A := Fix(H, C) with the Galois H-algebra B := Fix(G, C).

Proof : (i) By A.10 A ⊗_{R} B is semi-simple. If u ∈ A generates a G-normal basis of A over R and v ∈ B generates an H-normal basis of B over R then u ⊗ v generates a normal G × H-basis of A ⊗_{R} B over R. Hence A ⊗_{R} B is a Galois G × H-algebra.

(ii) Since G and H are normal in G × H we see from A.9 that A is a Galois Galgebra and B is a Galois H-algebra over R. Hence A⊗_{R}B is a Galois G×H-algebra over R, as shown in (i). The natural map A ⊗_{R} B → C is a map of G × H-algebras over R. The following lemma shows that this is an isomorphism.

A.12 Lemma: Suppose A|R is a Galois G-algebra and B|R is any G-

algebra. Let f : A → B be a homomorphism of G-algebras over R (i.e., a homomorphism of R-algebras which is also a homomorphism of G-modules). Then f is injective. If B too is a Galois G-algebra over R then f is an isomorphism.

Proof : It is enough to prove, for each e ∈ P (R), that the restriction eA → eB is injective. Thus we may suppose R = K to be a field.

In this case we know from the Structure Theorem A.5 that the primitive idempotents of A are permuted transitively by G. Hence the kernel of f cannot contain any primitive idempotent of A. For, if this would be the case then Ker(f) (which is stable under G) would contain all primitive idempotents, hence their sum 1, which gives a contradiction. Since A is semisimple we conclude Ker(f) = 0. Thus f is injective. If B too is a Galois G-algebra over K then both A and B have the same dimension over K and hence f(A) = B.

A.13 Tower of Galois algebras: Let 𝔤 and G be two finite groups. Let R′|R be a Galois 𝔤-algebra and A′|R′ be a Galois G-algebra. Suppose the action of 𝔤 on R′ is extended to an action of 𝔤 on the R-algebra A′ in such a way that as operator group on A′, 𝔤 commutes elementwise with G. (Thus A′|R can be viewed as G × 𝔤-algebra.) Put A := Fix(𝔤, A′). (Because of the imposed commuting condition A is a G-algebra over R.) Then we have:

(i) A′|R is a Galois G × 𝔤-algebra.

(ii) A|R is a Galois G-algebra.

(iii) The natural map A ⊗_{R} R′ → A′ is an isomorphism of G × 𝔤-algebras.

𝔤 A′ ≈ A ⊗_{R} R′ A = Fix(𝔤, A′) G A = Fix(𝔤, A′) |||||||||R  𝔤 R  R′ GR′ ||||||||| A′ ≈ A ⊗_{R} R′

Proof : (i) We again refer to A.3. So it suffices to show, for a given e ∈ P (R), that the G×𝔤-algebra eA′|eR is Galois; after changing notation we thus may assume that R = K is a field.

Since A′|R′ is a Galois G-algebra, it is a free R′-module of rank |G|. Similarly, [R′ : K] = |𝔤|. It follows

[A′ : K] = |G| · |𝔤| = |G × 𝔤| .  (A7)

Now the assertion follows from A.4, applied to the G × 𝔤-algebra A′|R; note that Fix(G × 𝔤, A′) = R = K.

(ii) and (iii) follow from (i), see Lemma A.12.

Hilbert’s Theorem 90: This theorem holds for arbitrary Galois G-algebras. However, we need it in this paper for a very special case only, namely for quadratic Galois G-algebras, which means that the group G is of order 2. In that case the statement and proof is rather trivial. For the convenience of the reader we shall present it here.

A.14 Lemma: Suppose A|R is a Galois G-algebra for a group G of

order 2, i.e., G = ⟨σ⟩ and σ^{2} = 1. Let a ∈ A^{×}. If a^{σ+1} = 1 then there exists b ∈ A^{×} such that a = b^{σ−1}. And conversely.

Proof : Using A.3 we may suppose that R = K is a field. If we put b := 1+a^{−1} then we compute b^{σ} = a · b using a^{σ} = a^{−1}. Hence if b is a unit in A then b is a solution of the problem.

If b is not a unit in A then b^{σ+1} is not a unit either; since b^{σ+1} is contained in the field K it follows b^{σ+1} = 0. But b^{σ+1} = ab^{2} and since a is a unit we have b^{2} = 0, hence b = 0 because A does not contain nilpotent elements. We conclude a = −1.

Thus if a = −1 then the above definition of b has to be modified. In that case we put b := u − u^{σ} where u ∈ A is chosen such that u, u^{σ} are linearly independent over K (normal basis). Then again, b^{σ} = −b = a · b. This time we can be sure that b is a unit: otherwise b = 0 (as above) which would imply that u and u^{σ} are linearly dependent over K.

The converse is directly verified.

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