- 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 ax
^{4}- by^{4}= 1'' by ``solutions of ax^{4}- by^{4}= 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 + (X
^{2}-m)Z[X]'' by ``X + (X^{2}-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
_{i}are 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 x
^{2}+ 1 \ne 0 by v^{2}+ 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}^{-i}theta by theta e_{chi}= B_{1,chi}^{-i}e_{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.

The statement of Prop. 1.5. is nonsense. What I (probably)
meant is that if fx^{2} + gy^{2} = hz^{2}
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_{p}G-module, the other
is to study Cl_{p}(K) as a Z_{p}G-module. The
proof using the Stickelberger element, however, does not work
over F_{p}G because of the p in the denominator. For a
corrected proof, see the ps-file of Chapter 11 that can be found
here.

- 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*, Ann. Math. (2)^{l}+ 1 = dy^{l}in a Galois field**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).