Irreduciblity of the polynomial $x^{p^n}-x+1$

Asked Sep 06 '12 at 05:56

Active Sep 06 '12 at 05:56

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Possible Duplicate:
Why $x^{p^n}-x+1$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$

What are the values of $n$ for which the polynomial $$f(x):=x^{p^n}-x+1$$is irreducible over the finite field $\mathbb{F}_p$ ?

I know that for $n=1$ the polynomial is irreducible. Are there any other $n$ for which $f(x)$ is irreducible ?

edited Apr 13 '17 at 12:21

Community

asked Sep 06 '12 at 05:56

pritam

2

This is a duplicate. Thomas Andrews gives an excellent answer there. – Jyrki Lahtonen Sep 06 '12 at 06:24
1

And see this related question for a specific example. – Jyrki Lahtonen Sep 06 '12 at 06:44
And this question with several answers (among which one by OP) that show why the polynomial is irreducible for $n=1$. – Marc van Leeuwen Sep 06 '12 at 07:30

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