Definition:Minimal Polynomial/Definition 1

Definition
Let $L / K$ be a field extension.

Let $\alpha \in L$ be algebraic over $K$.

The minimal polynomial of $\alpha$ over $K$ is the unique monic polynomial $f \in K[x]$ of smallest degree such that $f(\alpha) = 0$.

Also see

 * Equivalence of Definitions of Minimal Polynomial