Splitting field of an irreducible polynomial

**Bruno J.** · June 24th 2009, 01:38 PM

Suppose $K$ is the splitting field of an irreducible polynomial $p(x) \in \mathbb{Z}[x]$ . What is a general condition to have $[K:\mathbb{Q}] = \mbox{ deg }p(x)$ ? Is there a systematic way to test if that is the case, given $p(x)$ ?

**NonCommAlg** · June 25th 2009, 12:14 AM

Originally Posted by Bruno J.

Suppose $K$ is the splitting field of an irreducible polynomial $p(x) \in \mathbb{Z}[x]$ . What is a general condition to have $[K:\mathbb{Q}] = \mbox{ deg }p(x)$ ? Is there a systematic way to test if that is the case, given $p(x)$ ?

as far as i know there's no general condition in this case but if we replace $\mathbb{Q}$ with any finite field, then the result is always true. (the proof is quite easy!)

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