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.


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).

  • $\begingroup$ What set is F^p? Sorry, I've never seen that notation before. $\endgroup$ – James Norton Feb 12 '15 at 2:44
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    $\begingroup$ @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

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