TITLE: Field extension and irreducibility
QUESTION [5 upvotes]: Let $k$ a field, $P \in k[X]$ irreducible of degree $n \geq  2$, $K$ an extension field of $k$ with degree $m$ such as $\gcd(m,n) =1$. How can I show that $P$ stays irreducible over $K$ ? 
Thank for answers.

REPLY [4 votes]: This is basically the same as the other answers.
Let $x$ be a root of $P$ in some extension of $K$. Then $P$ is the minimal polynomial of $x$ over $k$. Let $d$ be the degree of $K(x)$ over $k$.
Contemplating the tower $k\subset K\subset K(x)$, we see that $m$ divides $d$.
Contemplating the tower $k\subset k(x)\subset K(x)$, we see that $n$ divides $d$.
Contemplating any of the above the towers, we see that $d$ doesn't exceed $mn$. 
As $m$ and $n$ are coprime, we have $d=mn$, and the first tower shows that $K(x)$ has degree $n$ over $K$. This implies that $P$ is irreducible over $K$.