Free pro-p extensions of number fields

by Kay Wingberg at Heidelberg

This paper concerns the problem of the existence of Galois extensions of algebraic number fields whose Galois groups are free pro-p groups. Let k be an algebraic number field and

F = G(N|k)

a free pro-p factor of the Galois group G(k(p)|k) of the maximal p-extension of k. Then N|k is a pro-p extension which is unramified outside p, i.e. F is a factor of the Galois group G_{Σ}(k) = G(k_{Σ}|k), where k_{Σ} is the maximal p-extension of k which is unramified outside the set Σ of primes of k lying above p or ∞. If

ρ(k) is the maximal possible rank

of such a free pro-p factor and assuming that Leopold’s conjecture holds for k and p, then 1 ≤ ρ(k) ≤ r_{2} + 1, where r_{2} denotes the number of complex places of k. Some examples are known where ρ(k) = r_{2} + 1 and there are also number fields with ρ(k) < r_{2} + 1, see [8]. If k is a global number field which contains the group μ_{2p} of 2p-th roots of unity and assuming that the generalized Greenberg conjecture holds (see §2), then Lannuzel/Nguyen Quang Do [2] and, independently, McCallum [3] proved that the case ρ(k) = r_{2} +1 only occurs when G_{Σ}(k) is itself a free pro-p group, i.e. k has only one prime above p and the pprimary part of its ideal class group is trivial. In this paper we give a short proof of this theorem assuming a weaker form of the Greenberg conjecture.

In general it seems to be difficult to find a free pro-p factor F of G_{Σ}(k) of rank bigger than 1. We will consider the case k = ℚ(ζ_{p}) and construct free factors under certain conditions.

1 Pro-p Operator Groups

Let p be a prime number. For a pro-p group G we denote its Frattini subgroup by G^{∗} = G^{p}[G, G]. For the cohomology groups of G with coeffients in ℤ/pℤ we often set H^{i}(G) = H^{i}(G, ℤ/pℤ). If A is an abelian group, then A^{∨} denotes its Pontryagin dual.

Let p be a prime number and let

\xymatrix{1 \ar[r] & G \ar[r] & G \ar[r] & Δ \ar[l]^{s} \ar[r] & 1 ,}

be a split exact sequence of profinite groups where G is a pro-p group and Δ is a finite group of order prime to p. Thus G is the semi-direct product of Δ by G and G is a pro-p-Δ operator group where the action of Δ on G is defined via the splitting s. Conversely, given a pro-p-Δ operator group G, we get a semi-direct product G = G ⋊ Δ where the action of Δ on G is the given one.

Let G(p) be the maximal pro-p quotient of G and let G_{Δ} be the maximal quotient of G with trivial Δ-action. Observe that G_{Δ} is well-defined. It can be shown ([7], proposition 1.1) that there is a canonical isomorphism

G_{Δ} ⟶^{∼} G(p) .

Furthermore, if Δ_{0} is a subgroup of Δ, then

H^{2}(G_{Δ_{0}}) ⟶^{inf} H^{2}(G)^{Δ_{0}}

is injective; in particular, if H^{2}(G)^{Δ_{0}} = 0, then G_{Δ_{0}} is a free pro-p group.

Proposition 1.1 Let p be an odd prime number and let Δ be a finite abelian group of exponent p − 1 with character group Δ^{∨}. Let G be a pro-p-Δ operator group, which is finitely generated as a pro-p group, and let

H^{2}(G) = ⨁_{χ∈Ω} H^{2}(G)^{χ}

be the decomposition into χ-eigenspaces of H^{2}(G) where Ω is the subset of Δ^{∨} given by the non-trivial eigenspaces. Assume that there exists a subgroup Δ_{0} of Δ such that

χ_{|_{Δ_{0}}} ≠ 1 for all χ ∈ Ω.

Then the maximal quotient E = G_{Δ_{0}} of G with trivial Δ_{0}-action is a free pro-p group of rank

r = ∑_{χ∈(Δ/Δ_{0})^{∨}} dim_{𝔽_{p}}(G/G^{∗})^{χ},

and there is an isomorphism

⨁_{χ∈(Δ/Δ_{0})^{∨}}(G^{ab})^{χ} ≅ E^{ab}

of ℤ_{p}[Δ]-modules.

Proof: As mentioned above we have H^{2}(G_{Δ_{0}}) ⊆ H^{2}(G)^{Δ_{0}}, and so

H^{2}(G_{Δ_{0}}) ⊆ ⨁_{ψ∈(Δ/Δ_{0})^{∨}} H^{2}(G)^{ψ} = ⨁_{ψ∈(Δ/Δ_{0})^{∨}} ( ⨁_{χ∈Ω} H^{2}(G)^{χ})^{ψ}.

From the exact sequence

0 ⟶(Δ/Δ_{0})^{∨} ⟶ Δ^{∨} ⟶ Δ_{0}^{∨} ⟶ 0

it follows that (Δ/Δ_{0})^{∨} ∩ Ω = ∅, and so we obtain H^{2}(E) = H^{2}(G_{Δ_{0}}) = 0, i.e. E is a free pro-p group. Since

E^{ab} = (G_{Δ_{0}})^{ab} = ⨁_{ψ∈(Δ/Δ_{0})^{∨}}(G^{ab})^{ψ}

the proposition is proved. □

2 The Greenberg Conjecture and Free Pro-p Extensions of Number Fields

We use the following notation:

p  is a prime number,
k  is a number field (not nessarily of finite degree over ℚ),
k_{∞}  is the cyclotomic ℤ_{p}-extension of k,
k̃  is the compositum of all ℤ_{p}-extensions of k,
Σ  is the set S_{p} ∪ S_{∞} of primes above p and archimedean primes,
k_{Σ}  is the maximal p-extension of k which is unramified outside Σ,
G_{Σ}(k) is the Galois group G(k_{Σ}|k) of k_{Σ} over k
Γ  is the Galois group G(k_{∞}|k),
L_{k}  is the maximal unramified p-extension of k,
L_{k}^{S_{p}}  is the maximal unramified p-extension of k,

which is completely decomposed at S_{p} .

If K|k is a Galois extension of number fields, then we denote the decomposition group of G(K|k) with respect to a prime 𝔭 by G_{𝔭}(K|k).

The groups of roots of unity of p-power order of k is denoted by μ(k)(p), and Cl(k)(p) and Cl_{S_{p}}(k)(p) is the p-primary part of the ideal class group and the S_{p}-ideal class group of k, respectively. Let r_{2} = r_{2}(k) be the number of complex places of k. Finally we set

X_{cs}(k) = G(L_{k}^{S_{p}}|k)^{ab} and X_{nr}(k) = G(L_{k}|k)^{ab}.

Let k be a number field of finite degree over ℚ, k^{(a)}|k a multiple ℤ_{p}-extension of rank a ≥ 1, i.e.

Γ^{(a)} = G(k^{(a)}|k) ≅ ℤ_{p}^{a},

and Λ = Λ_{(a)} the completed group ring ℤ_{p}[[Γ^{(a)}]]. The Λ-torsion submodule of a Λ-module M is denotes by T_{Λ}(M) and F_{Λ}(M) is the quotient M/T_{Λ}(M).

If Leopoldt’s conjecture for k and p holds, then the compositum k̃ of all ℤ_{p}- extensions of k is the unique multiple ℤ_{p}-extension k^{(r_{2}+1)} of rank r_{2} + 1. The following statement is called “generalized Greenberg conjecture”

GC(1) : X_{cs}(k̃) is a pseudo-null Λ-module,

and is due to Greenberg (stated for X_{nr}(k̃)) who generalized his earlier conjecture which asserts that for a totally real number field k the ℤ_{p}[[Γ]]-module X_{nr}(k_{∞}) is finite. A weaker form of the conjecture above is the following:

GC(2) : If X_{cs}(k̃)≠0, then it has a non-trivial pseudo-null Λ-submodule.

Lemma 2.1 Let k be a number field of finite degree over ℚ, F = G(N|k) a free pro-p factor group of G_{Σ}(k) of rank r_{2} + 1, and k_{∞} ⊆ k^{(a)} ⊆ N a multiple ℤ_{p}-extension of rank a. Then

T_{Λ}(G_{Σ}(k^{(a)})^{ab}) = G_{Σ}(N)/[G_{Σ}(N), G_{Σ}(k^{(a)})].

Proof: Let Λ = ℤ_{p}[[Γ^{(a)}]] and

φ : G_{Σ}(k^{(a)}) ↠ G(N|k^{(a)}).

Since G(N|k^{(a)}) is free, we obtain the exact sequence

0 ⟶ G_{Σ}(N)/[G_{Σ}(N), G_{Σ}(k^{(a)})] ⟶ G_{Σ}(k^{(a)})^{ab} ⟶^{φ^{ab}} G(N|k^{(a)})^{ab} ⟶ 0 .

The Λ-module G(N|k^{(a)})^{ab} is torsion-free of rank r_{2}, see [4] (5.6.6), and the Λrank of G_{Σ}(k^{(a)})^{ab} is also equal to r_{2} by [1] (4.3) and (5.4)(b) (observe that the weak Leopoldt conjecture holds since k_{∞} ⊆ k^{(a)}). Therefore the kernel of φ^{ab} is the Λ-torsion part T_{Λ}(G_{Σ}(k^{(a)})^{ab}) of G_{Σ}(k^{(a)})^{ab}. □

The following theorem is due to Lannuzel/Nguyen Quang Do [2] (assuming GC(1) and that all finite abelian p-extensions of k unramified outside p satisfy Leopoldt’s conjecture) and, independently, to McCallum [3] (for k = ℚ(μ_{p}) and assuming GC(1)).

Theorem 2.2 Let k be a number field of finite degree over ℚ containing the group μ_{2p}. Assume that Leopoldt’s conjecture for (k, p) and Greenberg’s conjecture GC(2) hold. Then the following assertions are equivalent:

(i) G_{Σ}(k) has a free pro-p factor group F of rank r_{2} + 1,

(ii) G_{Σ}(k) is a free pro-p group of rank r_{2} + 1,

(iii) #S_{p}(k) = 1 and Cl_{S_{p}}(k)(p) = 0.

Proof: For the well-known equivalence (ii)⇔(iii) see for example [4], (8.7.3). So we only have to prove the implication (i)⇒(ii).

Let F = G(N|k) be a free pro-p factor of G_{Σ}(k) of rank r_{2} + 1. Since Leopoldt conjecture holds, we have G(k̃|k) ≅ ℤ_{p}^{(r_{2}+1)} and k_{∞} ⊆ k̃ ⊆ N. Let Λ = ℤ_{p}[[ℤ_{p}^{(r_{2}+1)}]]. We consider the surjections

φ : G_{Σ}(k) ↠ G(N|k) and φ̃ : G_{Σ}(k̃) ↠ G(N|k̃).

By lemma (2.1) we have

T_{Λ}(G_{Σ}(k̃)^{ab}) = G_{Σ}(N)/[G_{Σ}(N), G_{Σ}(k̃)].

Let k_{0} = ℚ(μ_{2p}) and consider the abelian Galois group G(k̃_{0}|k_{0}) ≅ ℤ_{p}^{r+1}, where r = (p − 1)/2 if p > 2 and r = 1 otherwise. Its decomposition group G_{𝔭}(k̃_{0}|k_{0}) with respect to the unique prime 𝔭 above p has finite index, and so dim G_{𝔭}(k̃_{0}|k_{0}) = r + 1 ≥ 2. Since μ_{2p} ⊆ k, it follows that dim G_{𝔭}(k̃|k) ≥ 2 for all primes 𝔭 of k above p.

For a pro-p group G let I_{G} be the augmentation ideal of the completed group ring ℤ_{p}[[G]]. Setting E^{i}(−) = Ext_{Λ}^{i}(−, Λ) and using dim G_{𝔭}(k̃|k) ≥ 2 for all primes 𝔭|p, we obtain by Iwasawa theory an inclusion

X_{cs}(k̃)(−1) ↪ E^{1}(Y_{Σ})

with pseudo-null cokernel, where the Λ-module Y_{Σ} = I_{G_{Σ}(k)}/I_{G_{Σ}(k̃)}I_{G_{Σ}(k)} fits in an exact sequence

0 ⟶ G_{Σ}(k̃)^{ab} ⟶ Y_{Σ} ⟶ I ⟶ 0,

see [1] thm(5.4)(d), lemma(4.3) and [4](5.6.7); here I denotes the augmentation ideal of Λ. Analogously, we have an exact sequence

0 ⟶ G(N|k̃)^{ab} ⟶ Y_{F} ⟶ I ⟶ 0,

where

Y_{F} = I_{G(N|k)}/I_{G(N|k̃)}I_{G(N|k)} ≅ Λ^{r_{2}+1},

see [4](5.6.6) and recall that F = G(N|k) is a free pro-p group. We obtain a commutative and exact diagram

[figure: 0; G_{Σ}(k̃)^{ab}; Y_{Σ}; I; 0; 0; G(N|k̃)^{ab}; Λ^{r_{2}+1}; I; 0; ,]

and so an exact sequence

0 ⟶ T_{Λ}(G_{Σ}(k̃)^{ab}) ⟶ Y_{Σ} ⟶ Λ^{r_{2}+1} ⟶ 0.

It follows that

E^{1}(Y_{Σ}) ≅ E^{1}(T_{Λ}(G_{Σ}(k̃)^{ab})).

Therefore we get an inclusion

X_{cs}(k̃)(−1) ↪ E^{1}(T_{Λ}(G_{Σ}(k̃)^{ab}))

with pseudo-null cokernel, showing that X_{cs}(k̃) has no non-trivial pseudo-null Λ-submodule.

From our assumption GC(2) we get X_{cs}(k̃) = 0, and so

T_{Λ}(G_{Σ}(k̃)^{ab})^{∘} ∼ E^{1}(T_{Λ}(G_{Σ}(k̃)^{ab})) ∼ X_{cs}(k̃)(−1) = 0

(M^{∘} denotes the ℤ_{p}[[G(k̃|k)]]-module M with the inverse action of G(k̃|k)). It follows that

T_{Λ}(G_{Σ}(k̃)^{ab}) = 0,

since G_{Σ}(k̃)^{ab} has no non-trivial pseudo-null submodule, see [5] (4.2). Therefore G_{Σ}(N) = 1, i.e. k_{Σ} = N. This finishes the proof of the theorem. □

3 Free Pro-p Extensions of ℚ(ζ_{p})

We keep the notation of the preceding section. In the following we will construct free pro-p factors of G = G_{Σ}(k), where k = ℚ(ζ_{p}) and p is an odd prime number. The only method we have is to find a subextension k_{0} = k^{Δ_{0}} of k|ℚ, Δ_{0} ⊆ Δ = G(k|ℚ), such that G_{Σ}(k)_{Δ_{0}} = G((k_{0})_{Σ}|k_{0}) is a free pro-p-group.

Let ω be the Teichmüller character. We define the subsets Ω_{gen} and Ω_{rel} of characters of Δ = G(k|ℚ) by

Ω_{rel} = {ω^{i} ∈ Δ^{∨} | Cl(k)(p)^{ω^{1−i}} ≠ 0} and Ω_{gen} = {ω^{0}}∪Ω_{rel} ∪{ω^{i} ∈ Δ^{∨} | i odd}.

By Poitou-Tate duality and since Leopoldt’s conjecture holds for the abelian extension k|ℚ, we get

pG^{ab} ≅ H^{2}(G)^{∨}
≅ Hom(Cl(k)(p), μ_{p}) = ⨁_{ω^{i}∈Ω_{rel}}Hom(Cl(k)(p), μ_{p})^{ω^{i}}
≅ ⨁_{ω^{i}∈Ω_{rel}}(H^{2}(G)^{∨})^{ω^{i}} ≅ ⨁_{ω^{i}∈Ω_{rel}}(_{p}G^{ab})^{ω^{i}},

see [4] (8.6.13). Furthermore, since

G^{ab} ⊗ ℚ_{p} ≅ ℚ_{p} ⊕ ⨁_{i odd} ℚ_{p}[Δ]^{ω_{i}},

we have an 𝔽_{p}[Δ]-isomorphism

(G^{ab}/Tor_{ℤ_{p}}G^{ab})/p ≅ 𝔽_{p} ⊕ ⨁_{i odd} 𝔽_{p}[Δ]^{ω_{i}}.

From the exact sequence

0 ⟶ Tor_{ℤ_{p}}G^{ab} ⟶ G^{ab} ⟶ G^{ab}/Tor_{ℤ_{p}}G^{ab} ⟶ 0

and the fact that Tor_{ℤ_{p}}G^{ab}/p and _{p}G^{ab} are 𝔽_{p}[Δ]-isomorphic, it follows that

G_{Σ}(k)/G_{Σ}(k)^{∗} = ⨁_{ω^{i}∈Ω_{gen}}(G_{Σ}(k)/G_{Σ}(k)^{∗})^{ω^{i}}.

Since χ_{|_{Δ_{0}}} ≠ 1 for all χ ∈ Ω_{rel} if and only if (χ^{−1})_{|_{Δ_{0}}} ≠ 1 for all χ ∈ Ω_{rel} for a subgroup Δ_{0} ⊆ Δ and

H^{2}(G) = ⨁_{ω^{i}∈Ω_{rel}} H^{2}(G)^{ω^{−i}},

we get from proposition (1.1)

Theorem 3.1 Let p be an odd prime number, k = ℚ(ζ_{p}) and Δ = G(k|ℚ). Let Ω_{gen} = {ω^{0}} ∪ Ω_{rel} ∪ {ω^{i} ∈ Δ^{∨} | i odd} , Ω_{rel} = {ω^{i} ∈ Δ^{∨} | Cl(k)(p)^{ω^{1−i}} ≠ 0} . Assume that there exists a subgroup Δ_{0} of Δ such that

χ_{|_{Δ_{0}}} ≠ 1 for all χ ∈ Ω_{rel}, and let

Θ = Ω_{gen} ∩ (Δ/Δ_{0})^{∨}.

Then there exists a Δ-invariant surjection from G = G_{Σ}(ℚ(ζ_{p})) onto the free pro-p group E = G_{Δ_{0}} which induces an isomorphism

⨁_{χ∈Θ}(G^{ab})^{χ} ≅ E^{ab}

of ℤ_{p}[Δ]-modules. In particular,

rank E ≥ ∑_{χ∈Θ} dim_{𝔽_{p}}(G/G^{∗})^{χ} ≥ #Θ.

If Vandiver’s conjecture holds, i.e. Cl(k^{+})(p) = 0, then rank E = #Θ.

Remark: Since ω^{0} ∉ Ω_{rel}, it follows that ω^{0} ∈ Θ, and so E surjects onto Γ = G(ℚ(ζ_{p^{∞}})|ℚ(ζ_{p})), where ℚ(ζ_{p^{∞}}) is the cyclotomic ℤ_{p}-extension of ℚ(ζ_{p}).

In good cases we obtain small subgroups Δ_{0} such that χ_{|_{Δ_{0}}} ≠ 1 for all χ ∈ Ω_{rel}, and so large subsets Θ of Ω_{gen} with the properties as above. Let ℓ be a prime number and

w_{ℓ} = w_{ℓ}(Ω_{rel}) = \{ \begin{matrix}max\{v_{ℓ}(i) | 1 ≤ i < p − 1, ω^{i} ∈ Ω_{rel}\} , & if Ω_{rel} ≠ ∅, ℓ odd, \\ ∞, & if Ω_{rel} ≠ ∅, ℓ = 2, \\ −1, & otherwise. \end{matrix}

where v_{ℓ} is the ℓ-adic valuation. Consider the following set M(Ω_{rel}) of prime numbers ℓ dividing p − 1:

ℓ ∈ M(Ω_{rel}) ⇔ ℓ|p − 1 odd and w_{ℓ} < v_{ℓ}(p − 1) or ℓ = 2.

If Ω_{rel} = ∅, then M(Ω_{rel}) is the set of all prime divisors of p − 1. We identify Δ with

ℤ/(p − 1) = ⨁_{ℓ|p−1} ℤ/ℓ^{v_{ℓ}(p−1)} ,

and for ℓ ∈ M(Ω_{rel}) we define

Δ_{0}(ℓ) = \{ \begin{matrix}ℤ/ℓ^{w_{ℓ}+1} , & if Ω_{rel} ≠ ∅, ℓ odd, \\ Δ, & if Ω_{rel} ≠ ∅, ℓ = 2, \\ 1, & otherwise, \end{matrix}

and

Θ_{ℓ} = {ω^{0}} ∪ {ω^{k} | 1 ≤ k < p − 1 odd, v_{ℓ}(k) > w_{ℓ}} ⊆ (Δ/Δ_{0}(ℓ))^{∨}.

It follows that

χ_{|_{Δ_{0}(ℓ)}} ≠ 1 for all χ ∈ Ω_{rel}

and

#Θ_{ℓ} = 1 + \frac{p − 1}{2 · ℓ^{w_{ℓ}+1}} .

In particular, #Θ_{2} = 1 + (p − 1)/2 if Ω_{rel} = ∅, and #Θ_{2} = 1 otherwise. Interesting is the case when Ω_{rel} ≠ ∅ and M(Ω_{rel}) contains an odd prime number.

With the notation as above we obtain

Corollary 3.2 Let ℓ ∈ M(Ω_{rel}), where

Ω_{rel} = {ω^{i} ∈ Δ^{∨} | Cl(k)(p)^{ω^{1−i}} ≠ 0}.

Then there exists a Δ-invariant surjection from G = G_{Σ}(ℚ(ζ_{p})) onto a free pro-p group E, which surjects onto the cyclotomic ℤ_{p}-extension and has

rank E ≥ 1 + \frac{p − 1}{2 · ℓ^{w_{ℓ}+1}} .

In particular, if ℓ is odd, then E is non-abelian.

Remark: If p ≡ 3 mod 4, i.e. p − 1 = 2m, m odd, then G = G_{Σ}(ℚ(ζ_{p})) has a free non-abelian pro-p factor which surjects onto the cyclotomic ℤ_{p}-extension. Indeed, let Δ_{0} = 2Δ be the subgroup of Δ of order m. Then

H^{2}(G)^{Δ_{0}} ≅ (Cl(k)/p(−1))^{Δ_{0}} ≅ ((Cl(k)/p)^{ω^{1}} ⊕ (Cl(k)/p)^{ω^{m+1}})(−1)

((−1) denotes the (−1)-Tate-twist). Since the Bernoulli number B_{\frac{p+1}{2}} = B_{p−m} is not divisible by p (cf. [6] page 86), we have (Cl(k)/p)^{ω^{m}} = 0, and by Leopoldt’s Spiegelungsatz (see [6] thm 10.9) we get

dim_{𝔽_{p}}(Cl(k)/p)^{ω^{m+1}} ≤ dim_{𝔽_{p}}(Cl(k)/p)^{ω^{m}}.

Since also (Cl(k)/p)^{ω^{1}} = 0, it follows that H^{2}(G)^{Δ_{0}} = 0. The free factor G_{Δ_{0}} of G can be identify with the Galois group G_{Σ}(ℚ(\sqrt{−p})).

Example: Let k = ℚ(μ_{157}) and p = 157. Then

Ω_{rel}(k) = {ω^{62}, ω^{110}}

see [6] tables. Let Δ_{m} = ℤ/mℤ, m ≥ 1. Then

Δ = G(k|ℚ) = Δ_{156} = Δ_{4} ⊕ Δ_{3} ⊕ Δ_{13}

and the residues of i for ω^{i} ∈ Ω_{rel} are 62 = (2, 2, 10) and 110 = (2, 2, 6). It follows that

Θ_{3} = {ω^{0}} ∪ {ω^{j} | j odd and j ≡ 0 mod 3} ⊆ (Δ/Δ_{3})^{∨},
Θ_{13} = {ω^{0}} ∪ {ω^{j} | j odd and j ≡ 0 mod 13} ⊆ (Δ/Δ_{13})^{∨},

and (Δ/Δ_{i})^{∨} ∩ Ω_{rel} = ∅ for i = 3, 13. Therefore we have two G(k|ℚ)-invariant free pro-p factors F_{27} and F_{7} of G_{Σ}(k) of rank 27 and 7, respectively,

\begin{matrix}G_{Σ}(ℚ(μ_{157})) & ↠ & F_{27} ≅ G_{Σ}(ℚ(μ_{157})^{Δ_{3}}) , \\ G_{Σ}(ℚ(μ_{157})) & ↠ & F_{7} ≅ G_{Σ}(ℚ(μ_{157})^{Δ_{13}}) , \end{matrix}

such that there are G(k|ℚ)-invariant isomorphisms

\begin{aligned}F_{27}^{ab} & ≅ & ℤ_{p} ⊕ ℤ_{p}[Δ_{52}]^{−}, \\ F_{7}^{ab} & ≅ & ℤ_{p} ⊕ ℤ_{p}[Δ_{12}]^{−}, & p = 157. \end{aligned}

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[6] Washington, L.C. Introduction to Cyclotomic Fields. Springer 1982 (Second ed. 1997)

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Mathematisches Institut der Universität Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg Germany

e-mail: wingberg@mathi.uni-heidelberg.de