Polynomial of prime degree: being irreducible or having a root!

Question

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

Its actually quite simple, although no one has stated it. A proper divisor of $x^p-a$ must have a constant term of the form $\zeta^i(\sqrt[p]{a})^k$ now by taking an appropriate power you get a root in $K$. — Rene Schipperus, Dec 04 '14 at 13:41

score 0 · Accepted Answer · edited Apr 13 '17 at 12:20

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This question is a duplicate, and has already been asked (and answered) several times:

edited Apr 13 '17 at 12:20

Community

answered Dec 04 '14 at 09:22

Sean Elvidge

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This is not an answer. You can post all these links as a comment. – user26857 Nov 30 '15 at 14:11
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You're right. But you need to have 50 rep to make a comment to a question, which I didn't have at the time. – Sean Elvidge Nov 30 '15 at 14:57

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