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I'm interested in looking at the details of Gauss' method of determining the sign of the Gauss sum in his "Summatio Quarumdam Serierum Singularium", and I was wondering if anyone knew if there was an English or French translation available at all? I've tried searching on the internet and haven't found anything. Also, are there any modern presentations of the Gauss' method that you would recommend?

Thanks for your help.

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D M Bressoud, On the value of Gaussian sums, J Number Theory 13 (1981) 88–94, MR 82b:10046, "gives a new proof, closely paralleling that of Gauss...." – Gerry Myerson Nov 17 '10 at 4:58
Thanks-this looks like it will be really helpful – Lea M Nov 17 '10 at 5:28
up vote 9 down vote accepted

"The determination of Gauss sums" by Berndt and Evans (Bull. Amer. Math. Soc., Vol. 5, Number 2 (1981), 107-129.) contains an exposition of the original proof due to Gauss. It also includes a short historical outline of various classical and modern proofs due to Dirichlet, Cauchy and others.

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Thanks for linking to this article – Lea M Nov 17 '10 at 5:29
You're welcome. – Andrey Rekalo Nov 17 '10 at 5:43

there is a German translation: pages 463-495 in:

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(English + French)/2 = German? – Adam Hughes Dec 4 '10 at 1:04
actually, i would say that English $-$ French $\to$ German. – Suvrit Dec 4 '10 at 1:52
Unfortunately I only speak a tiny bit of German (I've just taken a very basic introductory course), but thanks for pointing out that there is a German translation. – Lea M Dec 7 '10 at 2:07

An English translation of Baumgart's exposition of Gauss's fourth proof can be found here.

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That looks great-thanks so much for providing the link. – Lea M Dec 7 '10 at 2:05

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