Let $p$ be a prime number. Prove that for any field $K$ and any $a \in K$, the polynomial $x^p−a$ is either irreducible, or has a root.
it doesn't seem hard, but i have no idea.
any hint is welcomed!
thank you
2026-03-28 21:06:31.1774731991
Polynomial of prime degree: being irreducible or having a root!
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This question is a duplicate, and has already been asked (and answered) several times:
Prove $f=x^p-a$ either irreducible or has a root. (arbitrary characteristic) (without using the field norm)
Irreducibility of a polynomial if it has no root (Capelli)
$x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible?