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.
Gauss' proof of the irreducibility of a cyclotomic polynomial
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algebraic-number-theory
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0You mean because it is usually proved using Eisenstein's criterion, and Eisenstein came later? – 2012-07-23
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0@GeoffRobinson Gauss proved it before 1800. Eisenstein was born in 1823. – 2012-07-23
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0Yes, 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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0By 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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0Yes, I think I have seen that mentioned too. But everyone knows the name of Eisenstein's criterion. – 2012-07-23
1 Answers
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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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0It's a modified version but there's a cite to the original version in the footnotes. – 2012-07-23
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0That's great. Thanks! – 2012-07-23