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Problem. Let K be a field and $a\in K$ such that $f(x)=x^n-a$ is irreducible. If $m\in \mathbb{N}$ divides n and $\alpha$ is a root of $f(x)$ in a extension field of K, find the minimal polynomial of $\alpha^m$.

Idea. I know that $x^p-a$ ($n=m\cdot p$) is a monic polynomial that has $\alpha$ like root but I don't know if it's irreducible or not.

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If you denote $g(x)=x^p-a$, then $f(x)=g(x^m)$. In particular, if $g(x)=h_1(x)\cdot h_2(x)$, then $f(x)=h_1(x^m)\cdot h_2(x^m)$, and since $f$ is irreducible, $h_1$ or $h_2$ has to be constant.

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We know $(\alpha^m)^p=\alpha^n=a$ thus it solves $$x^p-a=0$$ Moreover this is the minimal polynomial since every polynomial of lower degree would give a polynomial relation of degree lower than $n$ vanishing at $\alpha$, which is impossible because $n$ is the degree of $\alpha$.

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