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 lp = rx + sy = tz' by lp = 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 epsilonp' by epsilonq'.
• 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 Ok by OK [Brett Tangedal]
• p. 112, line 7: replace \xi \in Ok' by \xi \in Ok \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(\zetamp). [R. Auer]
• p. 130, Prop. 4.25.ii): replace Q(\zetam) by \Q(\zetamp)
• 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 aiare 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 epsiloni by ei
• 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 epsi by ei
• p. 377, lines 4, 8: replace echi by ei
• p. 378, lines 2, 4, 8: replace mCi by {\mathcal C}i
• p. 380, -12: replace echi theta = B1,chi-itheta by theta echi = B1,chi-iechi.
• 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, : replace Yamamoto by Yamamoto
• p. 422, : replace Minkowsi--Hasse'' by Minkowski--Hasse''
• p. 462, : replace Sierpinsky'' by Sierpinski'' [Kaplansky]
• p. 467, : 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 FpG-module, the other is to study Clp(K) as a ZpG-module. The proof using the Stickelberger element, however, does not work over FpG 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 cxl + 1 = dyl 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). 