Let $F$ be a field, $a$ an element of $F$ and $p$ prime.
How do I prove that

$f=X^p-a\in F[X]$ is irreducible iff $f$ has no root in $F$?

Honestly, I have no idea how to approach this. Maybe somebody can give me a push in the right direction?

  • 2
    $\begingroup$ Notice that one direction is easy : irreducible $\implies$ no root (in $F$). $\endgroup$ – Watson Mar 26 '16 at 13:31
  • $\begingroup$ Related : Prove $f=x^p-a$ either irreducible or has a root $\endgroup$ – Watson Mar 26 '16 at 13:33
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    $\begingroup$ Also note that this depends on $p$ being prime. For example, $X^4+1$ has no root in $\mathbb{R}$, but it is reducible. $\endgroup$ – hardmath Mar 26 '16 at 14:50