Degree of factors of the Artin–Schreier polynomial in $\mathbb{F}_q$.

Asked Mar 13 '24 at 18:33

Active Mar 13 '24 at 18:33

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Consider the field $\mathbb{F}_q$, where $q$ is a power of $p$, say $q=p^n$. Let $f=x^q-x-a\in\mathbb{F}_q[x]$, with $a\in\mathbb{F}_q$. I'm trying to determine the degree of the irreducible factors of $f$ in $\mathbb{F}_q$. Now I already solved this for $a=0$, since, $$x^q-x=x^{p^n}-x=\prod_{g \text{ monic, irreducible, deg}(g)\mid n}g,$$ but, for $a\neq 0$, I can't seem to make any progress at all.

Thanks in advance!

asked Mar 13 '24 at 18:33

Num2

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Related: Irreducible factors for $x^q-x-a$ in $\mathbb{F}_p$. – Shean Mar 13 '24 at 18:47
Maybe I'm confused but in the link you suggested the question is how $f$ factors in $\mathbb{F}_p$, but that is not what I'm asking. I'm interested what the degrees are of the factors in $\mathbb{F}_q$ (not $\mathbb{F}_p$, nor am I interested what the actual factors are, although of course, they could be of use). – Num2 Mar 13 '24 at 19:31
There is also this and many others linked to the thread Shean suggested. – Jyrki Lahtonen Mar 13 '24 at 19:47
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In other words, the irreducible factors in $\Bbb{F}_q[x]$ all have degree $p$. – Jyrki Lahtonen Mar 14 '24 at 04:52

Degree of factors of the Artin–Schreier polynomial in $\mathbb{F}_q$.

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