Let p be prime, and $K = {\bf F}_p (T)$ an extension of ${\bf F}_p$. How can i prove that the polynomial $x^p - T$ is irreducible in $K[x]$
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Needs correction – Wuestenfux Mar 15 '22 at 17:06
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Corrected now, thank you. – cesar col Mar 15 '22 at 17:08
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Algebraic or transcendental extension K – Wuestenfux Mar 15 '22 at 17:11
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Should it matter? My question is out of an example my teacher wrote, he just said the polynomial was irreducible in K, but he didnt proove the result nor specified if it was algebraic or trascendental. – cesar col Mar 15 '22 at 17:37
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Hint Since $F_p(T) = \operatorname{Quot}(F_p[T])$ and $x^p -T$ is a monic polynomial you may use Gauss‘ lemma to reduce to showing that $x^p-T$ is irreducible in $F_p[T][x]$, where it follows from Eisensteins criterion, because $T$ is prime.
Jonas Linssen
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This is the argument! I need to vote to close the question as a duplicate. Relatively experienced users can guess that this has been asked earlier. Don't worry too much, the upvote is mine. – Jyrki Lahtonen Mar 15 '22 at 18:12
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That’s fine. Since I am on mobile I am not good in finding links for duplicates and since the argument is short I thought Id simply restate it. – Jonas Linssen Mar 15 '22 at 18:15