Let $F$ be a field of arbitrary characteristic, $a\in F$, and $p$ a prime number. Show that $$f(X)=X^p-a$$ is irreducible in $F[X]$ if it has no root in $F$.
This answer to a related question mentions the result is due to Capelli.
I can prove the result if $F$ has characteristic $p$ as follows. Suppose $f$ is reducible: $f(X)=g(X)h(X)$ with $g(X)$ an irreducible factor of degree $m$, $1\le m
How would you show the result in other characteristics?