Reciprocity Laws

Errata to "Reciprocity Laws. From Euler to Eisenstein"

p. ix, line - 7: replace ``finite extension'' by ``finite normal extension'' [R. Auer]
p. x, line -12: replace ``that to look'' by ``that one should look''.
p. xi, line - 7 replace `solutions ax⁴ - by⁴ = 1'' by ``solutions of ax⁴ - by⁴ = 1'' [R. Auer]
p. xiii, line -5: replace ``presentation'' by ``presentation of'' [Brett Tangedal]
p. xiv, line 5: replace ``but it do'' by ``but I do'' [Brett Tangedal]
p. 15, line - 13: replace ``towards of'' by ``towards'' [R. Auer].
p. 25, line 22: replace `p = q = 1 mod 4' by `p = q = 3 mod 4' [Jim Long]
p. 28, Exer. 1.8. replace `l_p = rx + sy = tz' by `l_p = rx + sy + tz' [Jim Long]
p. 30, line 5: replace ``residcovered'' by ``rediscovered'' [I. Kaplansky]
p. 32, line -5: replace ``A, B, M, N in N'' by ``A, B, M, N in \Z''
p. 43, line -13: replace ``X + (X²-m)Z[X]'' by ``X + (X²-m)Q[X]'' [Brett Tangedal]
p. 44, lines 9-11: replace the `m' in e.g. `m < -4' by disc k [R. Auer]
p. 47, proof of Prop. 2.8: the exponent `sigma - 1' should be replaced by `1 - sigma'.
p. 60, line - 14: replace ``p-adic square'' by ``2-adic square'' [R. Auer]
p. 61, line - 3 replace ``implies that'' by ``implies'' [R. Auer]
p. 64: the entries of the Hilbert symbol in the produc formula should be `a,b'; instead of `m,n'. [Jim Long]
p. 70, line -11: delete the bracket after `reciprocity law'.
p. 71, line - 5 replace (p/q) = +1 by (p/q) = -1 [R. Chapman]
p. 73, line - 3 replace ``every factor'' by ``every odd factor'' [R. Chapman]
p. 74, Ex. 2.29: replace `epsilon_p' by `epsilon_q'.
p. 83, Prop. 3.4: replace \sqrt{p^*}\Z by \sqrt{p^*}\Z[X] [R. Auer]
p. 88, Cor. 3.10.vi): the capital {mathfrak P} should be {mathfrak p} [R. Auer]
p. 92, line - 11 insert a ``to' between `possible' and `improve' [R. Auer]
p. 92, Thm. 3.18 delete the `integers' before `smallest' [R. Auer]
p. 99, lines 8, 11: delete the factors (2/p). [P. Roquette]
p. 101, line 6: replace `g of p' by `g modulo p' [R. Auer]
p. 101, line 9: replace `b' by `a+1' [R. Auer]
p. 102, footnote line 1: replace `restricted us' by `restricted ourselves' [R. Auer]
p. 111, line -- 13 replace `an n-th root' by `a primitive n-th root' [R. Auer]
p. 112, Prop. 4.2.iii): replace O_k by O_K [Brett Tangedal]
p. 112, line 7: replace `\xi \in O_k' by `\xi \in O_k \setminus {mathfrak p}' [R. Auer]
p. 113, line - 2: the symbols (alpha/mathfrak p)_k should be replaced by (alpha/mathfrak p)_K.[R. Auer]
p. 126, line -3: the small pi must be replaced by a capital Pi [R. Auer]
p. 130, Prop. 4.25: replace `of {mathfrak p}' by ` of {mathfrak p} in Q(\zeta_mp). [R. Auer]
p. 130, Prop. 4.25.ii): replace Q(\zeta_m) by \Q(\zeta_mp)
p. 130, line 18 insert `symbol' after `of the Artin' [R. Auer]
p. 131, Prop. 4.28 replace `=fp' by `mathfrak p' [R. Auer]
p. 135, lines -13, -14: replace the blackboard F by a calligraphic F.
p. 136, line 10: the references [Wy1,Wy2] can be found on p. 42 [Brett Tangedal]
p. 158, line 9: replace ``desired equality (5.5)'' by ``desired equality (5.3)'' [R. Chapman]
p. 167, line -9: replace p|ABC by q|ABC.
p. 167, line -6: replace m = q by m=p.
p. 174: replace `Notes of Chapter 6.7,9' by `Notes of Chapters 6, 7 and 9'.
p. 190, table: replace 11 by -11.
p. 236, line -5: the coefficients a_iare in Z[1/2], not in Z.
p. 246, line -8: the brackets around phi(alpha/mu) have different size [R. Chapman]
p. 265, line 3: K(j(\sqrt{-5})) = K(\sqrt2) should be replaced by K(j(\sqrt{-5})) = K(\sqrt{-1}\,) [R. Auer]
p. 280, Ex. 8.19: there's a bracket ] missing after [kappa/pi. [R. Chapman]
p. 294, Prop. 9.5.: replace the congruence `c = (p-3)/4 mod 4' by `c = -(p+1)/4 mod 4'.
p. 300, lines -8 to - 6: the z in phi(*/z) should be replaced by pi [R. Auer]
p. 302, line -14: replace `did no' by `did not' [Brett Tangedal]
p. 312, line 3: replace `Theorem 9.18' by `Theorem 9.19' [R. Auer]
p. 315, Ex. 9.9: add a bracket ) after `computer'.
p. 318, line 16: replace x² + 1 \ne 0 by v² + 1 \ne 0 [R. Auer]
p. 319, line 3: insert `(w,v) = ' in front of (\pm i,0), (0,\pm i) [R. Auer]
p. 321, line - 15: replace `m-the' by `m-th' [R. Auer]
p. 330, line 3: replace 2n <= 50 by 2n <= 32.
p. 351, Cartier: add a bracket after `fonction zeta'
p. 355, line - 4: replace `J.F. Felipe' by `J.F. Voloch' [R. Chapman]
p. 377, line 2: replace epsilon_i by e_i
p. 372, line 12: add space between element and attached [R. Chapman]
p. 372, line 15: the condition alpha < m/2 should be a subscript to the sum [R. Chapman]
p. 376, lines -3, -2, and p. 377, line 2: replace eps_i by e_i
p. 377, lines 4, 8: replace e_chi by e_i
p. 378, lines 2, 4, 8: replace mC_i by {\mathcal C}_i
p. 380, -12: replace e_chi theta = B_1,chi^-itheta by theta e_chi = B_1,chi^-ie_chi.
p. 394, line -10: replace `annihilate' by `annihilates'
p. 406, Kleboth: replace `Gle-ichung' by `Glei-chung'.
p. 415, Teege 2: replace 1921 by 1925
p. 418, 7th problem: the `Eisenstein sums' there are actually elliptic Gauss sums.

p. 444, [424]: replace Yamamoto by Yamamoto

p. 422, [55]: replace ``Minkowsi--Hasse'' by ``Minkowski--Hasse''

p. 462, [735]: replace ``Sierpinsky'' by ``Sierpinski'' [Kaplansky]

p. 467, [808]: replace 1895/86 by 1895/96

The statement of Prop. 1.5. is nonsense. What I (probably) meant is that if fx² + gy² = hz² has integral solutions, then gh (hf, -fg) are quadratic residues modulo every prime divisor of f (g, h). [R. Chapman]

Lemma 3.13: The discussion involving the index m ended up in the wrong part of the proof; see the tex/dvi/ps files for a corrected version. [R. Chapman]

My ``proof'' of Herbrand's Theorem in Chapter 11 is nonsense. The confusion arose because I mixed two possible descriptions of the theorem: one way of looking at it is by considering Cl(K)/Cl(K)^p as an F_pG-module, the other is to study Cl_p(K) as a Z_pG-module. The proof using the Stickelberger element, however, does not work over F_pG because of the p in the denominator. For a corrected proof, see the ps-file of Chapter 11 that can be found here.

Additions to "Reciprocity Laws. From Euler to Eisenstein"

Page 20: Teege's first attempt at filling the gap in Legendre's proof was incomplete: see his correction in Richtigstellung eines früheren Beweises für den Satz, daß es für jede Primzahl p von der Form 4n+1 unendlich viele Primzahlen von der Form 4n+3 gibt, von denen p quadratischer Nichtrest ist und Herleitung des Satzes, daß mindestens eine unter ihnen kleiner als p ist, Hamb. Mitt. 6 (1924), 100-106.
Page 110, Reference [Te1]: The title of Teege's dissertation is Über die (p-1)/2-gliedrigen Gaussischen Perioden in der Lehre von der Kreisteilung und ihre Beziehungen zu anderen Teilen der höheren Arithmetik.
Page 138: the Davenport-Hasse theorem for Jacobi sums (Cor. 4.33) is actually due to H.H. Mitchell [ On the congruence cx^l + 1 = dy^l in a Galois field, Ann. Math. (2) 18 (1917), 120-131], who expressed the result in terms of cyclotomic numbers.
Page 141: Schwering discusses the quintic power residue character of 2, 3 and 5 as well as the quintic period equation.
Page 170: Another proof of the quadratic reciprocity law in Z[i] based on Hilbert's genus theory can be found in S. Kuroda [ Über den Dirichletschen Körper, J. Fac. Sci. Univ. Tokyo, Sect. I 4 (1943), 383-406].
Page 173: Estes and Pall give another proof of Burde's reciprocity law.
Page 200: The version of the quartic reciprocity law (cf. Exercise 6.17) credited to unpublished papers of Gauss and Artin already occurs in Busche.
J. Sochocki, Bestimmung der constanten Factoren in den Formeln für die lineare Transformation der Thetafunctionen. Die Gauss'schen Summen und das Reciprocitätsgesetz der Legendre'schen Symbole, Par. Denkschrift, 1878, gives another proof of the quadratic reciprocity law using theta functions. See JFM
Brett Tangedal has given a proof of the quadratic reciprocity law for Jacobi symbols based on Eisenstein's original proof.
Robin Chapman has given a new proof of the quadratic reciprocity law by generalizing Nakash's proof that (5/p) = (p/5).