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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**