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Assume that $F$ is a field and $\operatorname{char}(F)=p$. Let $a$ be an element in $F$ without $p$th root, then the polynomial $$x^{p^n}-a$$ is irreducible and inseparable over $F$ for all $n$.

I have proved the inseparable part by considering the derivative of the polynomial, but I'm having trouble with the irreducible part. Any help?

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up vote 8 down vote accepted

Let $\alpha$ be a $p^n$-th root of $a$ in some extension of $F$. Then $x^{p^n} - a = (x - \alpha)^{p^n}$. If you have a non-trivial monic factor of this polynomial in $F[x]$, then it is of the form $(x - \alpha)^k$ for some $0 < k < p^n$. Can you get a contradiction out of the coefficients of this polynomial? It might be good to write $k = p^rs$ with $s$ not divisible by $p$.

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