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Let $n$ be a prime number. How can I show that the polynomial $f(x) = x^{n-1} + x^{n-2} + x^{n-3}+ \cdots + x+ 1$ is irreducible over any finite field?

Michael Hardy
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Kirill Bespalov
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  • Any field is not right. Take for example the complex numbers. – André Nicolas Dec 20 '14 at 00:58
  • It is not irreducible over $\mathbb{Z}_n$, because $1+\mathbb{Z}$ is a root. –  Dec 20 '14 at 01:02
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    I'm sorry, but I made a mistake. Of course I meant finite fields. – Kirill Bespalov Dec 20 '14 at 01:02
  • Indeed, this is not true. Take Questioner's example. – Alex Wertheim Dec 20 '14 at 01:04
  • Possible duplicates: http://math.stackexchange.com/questions/578642/irreducible-polynomial-over-a-finite-field or http://math.stackexchange.com/questions/433779/factoring-polynomials-of-the-form-1x-cdots-xp-1-in-finite-field ? – epimorphic Dec 20 '14 at 01:05
  • It's not true over all finite fields, either: if $n > 2$, take a finite field and adjoin all roots of $f$, yielding a (possibly larger) finite field in which $f$ isn't irreducible. – Daniel Hast Dec 20 '14 at 01:20
  • $;f(x)=x^{p-1}+x^{p-2}+\ldots+x+1;$ has $;1;$ as a root in $;\Bbb F_p;$ and is thus reducible there. – Timbuc Dec 20 '14 at 01:54

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