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I tried this problem for a while, but didn't see the application of Eisenstein's irreducibility criterion here. All the coefficients, including the leading coefficient, are equal to 1.

p is a prime number.

The polynomial is $f(x) = x^{p-1} + x^{p-2} + \cdots + x + 1$

How do I proceed? Many thanks.

user21820
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coolcheetah
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    Hint : try to show that $f(x+1)$ is irreducible and then deduce that $f(x)$ must be irreducible. Another way to do this is to realize that this is the cyclotomic polynomial $\Phi_p$ and hence irreducible (but you need to know that cyclotomic polynomials are always irreducible over $\mathbb{Q}$). – Clément Guérin Jun 17 '15 at 06:24
  • ClémentGuérin's answer is a great way to proceed. Might also be worth noting that this is the $p^{th}$ cyclotomic polynomial https://en.wikipedia.org/wiki/Cyclotomic_polynomial. – CPM Jun 17 '15 at 06:27
  • Click on "Edit" to see how I used LaTeX to write the mathematical expressions. You can find a LaTeX guide on Meta. – user21820 Jun 17 '15 at 06:38
  • http://math.stackexchange.com/questions/215042/irreducibility-of-xp-1-cdots-x1 – user26857 Jun 17 '15 at 07:44
  • http://math.stackexchange.com/questions/87609/eisenstein-criterion-with-a-twist – user26857 Jun 17 '15 at 07:44
  • http://math.stackexchange.com/questions/588310/xp-1-x2-x-1-is-irreducible-using-eisensteins-criterion – user26857 Jun 17 '15 at 07:45
  • http://math.stackexchange.com/questions/1277577/proof-that-fracxp-1x-1-1-x-dots-xp-1-is-irreducible – user26857 Jun 17 '15 at 07:45
  • http://math.stackexchange.com/questions/879119/if-p-is-a-prime-prove-that-xp-1-xp-2-xp-3-cdots-x1-is-irred – user26857 Jun 17 '15 at 07:45

1 Answers1

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$f(x) = \frac{x^p - 1}{x-1}$. [This division is valid in the fraction field of polynomials.]

Thus $f(t+1) = \frac{(t+1)^p - 1}{t}$.

Eisenstein's criterion applies and so $f(t+1)$ is irreducible in $\mathbb{Z}[t]$.

Now $f(x)$ cannot be reducible in $\mathbb{Z}[x]$ otherwise you could substitute $x$ by $(t+1)$ in a non-trivial factorization of $f(x)$ to get a non-trivial factorization of $f(t+1)$.

user21820
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