1
$\begingroup$

$$x^n-a$$

So $n$ is any integer greater than 1, and $a$ is any integer. $a$ being any integer is where I am running into trouble. I have already shown and worked out a proof for this being irreducible when a is prime. Now I am just working through examples, trying to figure out a pattern.

The exercise is asking for application of Eisenstein's Criterion only, so I am assuming that the value of n should have no effect on the irreducibility.

$\endgroup$
3
$\begingroup$

Theorem $\ $ Suppose $\,F\,$ is a field and $\:a\in F\:$ and $\:0 < n\in\mathbb Z.\ $ Then

$\ \ \ x^n - a\ $ is irreducible over $\:F \iff a \not\in F^{\large p}\:$ for all primes $\:p\mid n,\:$ and $\ a\not\in -4\:F^4$ when $\: 4\mid n $

For a proof see e.g. Karpilovsky, Topics in Field Theory, Theorem 8.1.6, or see Lang's Algebra (Galois Theory).

$\endgroup$
  • $\begingroup$ What set is F^p? Sorry, I've never seen that notation before. $\endgroup$ – James Norton Feb 12 '15 at 2:44
  • 1
    $\begingroup$ @James The $\,p$'th powers of elements of $\,F,\,$ i.e. $\,\{c^p\,:\, c\in F\}\ \ $ $\endgroup$ – Bill Dubuque Feb 12 '15 at 2:45
  • $\begingroup$ @JyrkiLahtonen There's an entry on my lengthy todo list to post a proof and clean up the dupes, but it is far from the top (highest priority) at the moment. $\endgroup$ – Bill Dubuque Aug 11 '19 at 15:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.