Prove $f=x^p-a$ either irreducible or has a root. (arbitrary characteristic) (without using the field norm)

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  • Irreducibility of a polynomial if it has no root (Capelli) [duplicate]

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Starting where you have left off, if $f=gh$, then $g$ must be a product of terms of the form $x-\zeta^i\beta$, where my $\beta$ is your $\alpha^{1/p}$ (which I think is supposed to be $a^{1/p}$). Given $i$ and $j$ where $x-\zeta^i\beta$ is a factor of $g$ and $x-\zeta^j\beta$ isn’t, there’s an automorphism taking $\zeta^i$ to $\zeta^j$. The automorphism fixes $g$, since the coefficients of $g$ are in $K$, but it doesn’t fix the factors of $g$, contradiction, since we have unique factorization of polynomials over a field.