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Let $f(x)$ be an irreducible polynomial over $F[X]$, with $char(F)=p$ prime number. We know that $ \exists t \in \mathbb{N} | f(x)=g((x^p)^t)$. We shall prove that $g(x)$ is irreducible and separable.

I know a Proposition which states that if $f(x)$ is irreducible it is equivalent to be in the form $f(x)=h(x^p)$ with $char(F)=p\ne 0$ and have multiple roots. But I can't figure out how to fruitfully apply it.

Thanks in advance!

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up vote 0 down vote accepted

Look at this link: this is well-known theorem in Galois/Field Theory. See proposition 2.5 and above.

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Thank you! Also for the book, it will definitely help! – GivAlz Oct 29 '12 at 19:22

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