Errata, Remarks, and Reviews for 

Bruce  C.  Berndt,  Ronald  J.  Evans, and  Kenneth  S.  Williams
Wiley-Interscience, N.Y., 1998

Table of Contents


(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 tex2html_wrap_inline30 
and tex2html_wrap_inline32  on  p. 238
are distinct. Thus our proof only verifies the theorem under
the additional restriction that tex2html_wrap_inline34 
and tex2html_wrap_inline36  have coprime norms.
In order to complete the proof, we must verify that the cubic
residue symbols  tex2html_wrap_inline38  and  
tex2html_wrap_inline40  are equal, where tex2html_wrap_inline42 is a 
primary Eisenstein prime whose norm p is congruent to 1 
modulo 3,  and tex2html_wrap_inline48  is the complex conjugate of 
tex2html_wrap_inline42.   We can
give a short proof of this as follows, using what was already
proved so far.  First, writing tex2html_wrap_inline52,  we have

Next, reversing the roles of  tex2html_wrap_inline42  
and  tex2html_wrap_inline48,  we obtain
The symbols on the right of these equalities are equal,
since tex2html_wrap_inline62  is congruent to tex2html_wrap_inline64  
modulo tex2html_wrap_inline66. 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  tex2html_wrap_inline66  is a cubic
residue modulo tex2html_wrap_inline42. But as was pointed out on p. 239,  
where  is defined (up to sign) by the quadratic form
tex2html_wrap_inline76  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  tex2html_wrap_inline88.  
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
tex2html_wrap_inline90  by  tex2html_wrap_inline92.   
On p. 444,  in formula (13.2.6), replace
tex2html_wrap_inline94  by  tex2html_wrap_inline96.  
On p. 468,  three and four lines after
(14.1.1), the nonzero ideals of  tex2html_wrap_inline98  and the 
nonzero elements
tex2html_wrap_inline34  in  tex2html_wrap_inline98 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 
by  "The inner sum on   vanishes".

(Note that the sum on   y  vanishes 
unless   =  1 + pt
with  tex2html_wrap_inline122, 
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 ].


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