Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
You mean because it is usually proved using Eisenstein's criterion, and Eisenstein came later? – Geoff Robinson Jul 23 '12 at 8:45
@GeoffRobinson Gauss proved it before 1800. Eisenstein was born in 1823. – Makoto Kato Jul 23 '12 at 9:00
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?) – Geoff Robinson Jul 23 '12 at 10:31
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. – Makoto Kato Jul 23 '12 at 11:49
Yes, I think I have seen that mentioned too. But everyone knows the name of Eisenstein's criterion. – Geoff Robinson Jul 23 '12 at 12:12
up vote 7 down vote accepted

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

share|cite|improve this answer
It's a modified version but there's a cite to the original version in the footnotes. – daniel Jul 23 '12 at 9:14
That's great. Thanks! – Makoto Kato Jul 23 '12 at 9:31

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.