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Let $F$ be a field with char$F = p$. Prove that a polynomial of the form $x^p−x−a \in F[x]$ is either irreducible or splits in $F$.

I've seen a few different ways of proving this with $x^p-a$. I know that $a \in F$ being a root of $F(x)$ means $a^p=a$, so $x^p−a=(x−a)^p$. But I don't think this property works for $x^p-x-a$.

Tim
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1 Answers1

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This is an Artin-Schreier polynomial. If $\alpha$ is a root, the other roots are $\alpha+k$ for $k\in\Bbb F_p$. If $\alpha\notin F$ then for some nonzero $k\in\Bbb F_p$, $\alpha+k$ is a conjugate of $\alpha$ over $F$. Then $\alpha+2k$ is a conjugate of $\alpha+k$ and so also a conjugate of $\alpha$, etc. Then all $\alpha+mk$ are conjugates of $\alpha$, and these are all the roots. Therefore $x^p-x-a$ is irreducible in this case.

Angina Seng
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  • So does it split when $a=0$? Because this seems to only work for $a$ nonzero? – Tim May 02 '18 at 17:02
  • Of course it splits when $a=0$, but also in other cases when $F\ne\Bbb F_p$. For instance, $X^2-X-1$ splits over $\Bbb F_4$. – Lubin May 02 '18 at 18:15