GAUSS AND JACOBI SUMS

**
Errata**

(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 aredistinct. Thus our proof only verifies the theorem under the additional restriction that and have coprimenorms.In order to complete the proof, we must verify that the cubic residue symbols and are equal, where is a primary Eisenstein prime whose normpis 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, whereris defined (up to sign) by the quadratic form Thus we obtain the interesting application thatris always a cubic residue ofp. For example,r= 7 is a cubic residue of the primep =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 eachtby . 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 tok. On p. 470, in each of the two lines above (14.2.2), replaceMby (k-1)M.(3)On p. 388, line 6, replaceabby 2ab.(4)On p. 41, line 13, replace "The inner sum onyvanishes" by "The inner sum onxvanishes". (Note that the sum onyvanishes unlessz= 1 +ptwith , while for each such nonzerot, the sum onxvanishes because the subsum over thosexcongruent toc(modp) vanishes for each fixedc.)(5)In Problem 22 on p. 336, we takekto beq-1.(6) Grammatical mistakes:On p. 531, line 2, capitalize the wordsums.Throughout the text, the possessive formGauss'should perhaps be replaced byGauss's.

**
Notes**

Research Problem 6on p. 496 has been solved forg(12) by R. J. Evans [Gauss sums of orders six and twelve, Can. Math. Bull.44(2001), 22-26 ]. (The problem forg(8) is still open.) Also in this paper is a new evaluation ofg(6) which is more elegant than that given on p. 156 in Theorem 4.1.4 of the book.Research Problems 9, 10on 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 11on p. 497 has been solved by R. J. Evans, [Nonexistence of twentieth power residue difference sets, Acta Arith.84(1999), 397-402 ].Research Problem 25on p. 498 has been solved by R. J. Evans [Classical congruences for parameters in binary quadratic forms, Finite Fields and their Applications7(2001), 110-124.]Research Problem 26on 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 ].