$E=F[\alpha]$, where $\alpha$ is a root of $x^p-x-c$.

Asked Jul 23 '21 at 20:13

Active Jul 23 '21 at 20:15

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Let $F$ be a field of characteristic $p>0$, and let $E/F$ be a Galois extension with cyclic galois group $\langle \sigma \rangle$ of order $p$.

I want to prove that $E=F(\alpha)$, where $\alpha$ is a root of an irreducible polynomial in $F[x]$ of the form $x^p-x-c$. I know that this polynomial is irreducible if and only it don't has roots on F. Any idea about how I can prove this?

It is an exercice of Rotman, Galois theory

edited Jul 23 '21 at 20:15

HallaSurvivor

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asked Jul 23 '21 at 20:13

pmtm

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Does this answer your question? So-called Artin-Schreier Extension. Even if it doesn't, the keyword "Artin-Schreier Extension" will definitely be helpful ^_^ – HallaSurvivor Jul 23 '21 at 20:17
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Yes, thankss :) – pmtm Jul 23 '21 at 20:20

$E=F[\alpha]$, where $\alpha$ is a root of $x^p-x-c$.

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