Let $E/K$ be a field extension and let $\alpha \in E$ be algebraic over $K$. Let $f \in K[x]$.
My question is, if I have the following:
- $f$ is monic
- $\deg(f) = [E : K]$
Can I then conclude that
- $f$ is the minimal polynomial for $\alpha$ over $K$?
Then I don't have to prove that $f$ is irreducible over $K$.