Splitting field of an irreducible polynomial

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

Quote:

---

Originally Posted by Bruno J.

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

---

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