Re: [HM] Gauss and the Jacobi symbol
William C Waterhouse (wcw@math.psu.edu)
Mon, 27 Sep 1999 17:55:58 -0400 (EDT)
On Mon, 27 Sep 1999, Franz Lemmermeyer <lemmerm@mpim-bonn.mpg.de>
wrote:
>...
> while reading my newly acquired Disquisitiones arithmeticae
> I stumbled across something in Gauss's first proof of the
> quadratic reciprocity law that seems odd to me: after stating
> the theorem in article 131, he deduces the reciprocity law
> for the Jacobi symbol in 133 (in different language: Gauss
> didn't have the Jacobi symbol). Nothing surprising here,
> but in article 132 he states some theorems that, if I
> understand them correctly, are false. The first one reads
> 9. a R A ===> A R a,
> where m R n is short for "there exists an integer x such that
> m \equiv x^2 \bmod n", and where a and A are _integers_
> (not necessarily prime) of the form 4r+1.
> Now clearly 21 R 5 since 21 = 1^2 mod 5, but 5 N 21
> (since 5 N 3) so we have a counterexample. The proof that
> Gauss gives for 9 is correct for _prime_ values of a and
> holds for arbitrary integers A.
>
> Although Gauss, e.g. in article 108, uses "number" in the
> sense of "prime number", in article 132 he says expressis
> verbis that "This table [in art. 131] includes all the
> cases when two prime numbers are compared; what follows
> pertains to any numbers, but the demonstration of these
> is less obvious."
>...
As one would guess, of course, there is no error. The notation
is explained just after the statement of quadratic reciprocity
in article 131:
"To indicate our reasoning as briefly as possible, we will
denote prime numbers of the form 4n+1 by the letters a, a', a'', etc.
and prime numbers of the form 4n+3 by the letters b, b', b'', etc.;
any numbers of the form 4n+1 will be denoted by A, A', A'', etc;
any numbers of the form 4n+3 by B, B', B'', etc...."
The capital letters introduced there just aren't needed until art. 132.
The notation continues in effect through art. 134.
William C. Waterhouse
Penn State