(1) We are very grateful to Richard A. Mollin for pointing out that our proof of the cubic reciprocity law (Theorem 8.1.7) is only valid when the primes and on p. 238 are distinct. Thus our proof only verifies the theorem under the additional restriction that and have coprime norms. In order to complete the proof, we must verify that the cubic residue symbols and are equal, where is a primary Eisenstein prime whose norm p is congruent to 1 modulo 3, and is the complex conjugate of . We can give a short proof of this as follows, using what was already proved so far. First, writing , we have Next, reversing the roles of and , we obtain The symbols on the right of these equalities are equal, since is congruent to modulo . Hence the symbols on the left are equal, and the proof is complete. Remark: Since the symbols on the left of these equalities are complex conjugates of each other by Proposition 8.1.3, it follows that they each equal 1. Thus is a cubic residue modulo . But as was pointed out on p. 239, where r is defined (up to sign) by the quadratic form Thus we obtain the interesting application that r is always a cubic residue of p. For example, r = 7 is a cubic residue of the prime p = 73. (2) We are very grateful to Charles Helou for pointing out the following list of mistakes. On p. 271, in Theorem 9.2.6, replace each t by . On p. 362, before formula (11.6.1), the reference should be to Theorem 2.1.3(a) instead of Theorem 1.1.4(a). On p. 365, after (11.6.4), replace by . On p. 444, in formula (13.2.6), replace by . On p. 468, three and four lines after (14.1.1), the nonzero ideals of and the nonzero elements in must be taken relatively prime to k. On p. 470, in each of the two lines above (14.2.2), replace M by (k-1)M. (3) On p. 388, line 6, replace ab by 2ab. (4) On p. 41, line 13, replace "The inner sum on y vanishes" by "The inner sum on x vanishes". (Note that the sum on y vanishes unless z = 1 + pt with , while for each such nonzero t , the sum on x vanishes because the subsum over those x congruent to c (mod p) vanishes for each fixed c.) (5) In Problem 22 on p. 336, we take k to be q-1. (6) Grammatical mistakes: On p. 531, line 2, capitalize the word sums. Throughout the text, the possessive form Gauss' should perhaps be replaced by Gauss's.
Research Problem 6 on p. 496 has been solved for g(12) by R. J. Evans [Gauss sums of orders six and twelve, Can. Math. Bull. 44 (2001), 22-26 ]. (The problem for g(8) is still open.) Also in this paper is a new evaluation of g(6) which is more elegant than that given on p. 156 in Theorem 4.1.4 of the book. Research Problems 9, 10 on pp. 496--497 have been solved by R. J. Evans, M. Minei, and B. Yee [Incomplete higher order Gauss sums, J. Math. Anal. Appl. 281 (2003), 454-476]. Research Problem 11 on p. 497 has been solved by R. J. Evans, [Nonexistence of twentieth power residue difference sets, Acta Arith. 84 (1999), 397-402 ]. Research Problem 25 on p. 498 has been solved by R. J. Evans [Classical congruences for parameters in binary quadratic forms, Finite Fields and their Applications 7 (2001), 110-124.] Research Problem 26 on p. 498 has been solved by R. J. Evans [Extensions of classical congruences for parameters in binary quadratic forms, Acta Arith. 100 (2001), 349-364 ].