If [itex]K[/itex] is a field of characteristic [itex]p[/itex], and there exists an element [itex]a \in K[/itex](adsbygoogle = window.adsbygoogle || []).push({});

which is not a [itex]p[/itex]th power (i.e. the Frobenius endomorphism is not

surjective), then I am told we can show [itex]x^p - a[/itex] is an irreducible polynomial

(and since it is not separable our field is imperfect). I see that

[itex]x^p - a[/itex] has no roots in [itex]K[/itex], but how do we know that there does not exist

any factorization of [itex]x^p -a[/itex] into factors of lesser degree?

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# Why is x^p - a irreducible over a field of characteristic p?

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