# Math Help - Splitting field of an irreducible polynomial

1. ## Splitting field of an irreducible polynomial

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 $p(x)$?

2. 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!)