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Let $l$ be an odd prime number. Let $f(X) = 1 + X + ... + X^{l-1} \in \mathbb{Z}[X]$. Probably Gauss was the first man who proved that $f(X)$ is irreducible. I wonder how he proved it.

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    You mean because it is usually proved using Eisenstein's criterion, and Eisenstein came later?2012-07-23
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    @GeoffRobinson Gauss proved it before 1800. Eisenstein was born in 1823.2012-07-23
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    Yes, thanks. I did not know the precise dates, but I knew Eisenstein was later (maybe even a student of Gauss, or maybe of Riemann?)2012-07-23
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    By the way, I heard that someone already published the Eisenstein's theorem before him. So that the name of the theorem may not be appropriate.2012-07-23
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    Yes, I think I have seen that mentioned too. But everyone knows the name of Eisenstein's criterion.2012-07-23

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The first proof presented here is a proof by Gauss. The original is in Gauss' magnum opus Disquisitiones Arithmeticae.

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    It's a modified version but there's a cite to the original version in the footnotes.2012-07-23
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    That's great. Thanks!2012-07-23