Definition:Minimal Polynomial/Definition 2

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 irreducible, monic polynomial $f \in K \sqbrk x$ such that $\map f \alpha = 0$.

Also see

 * Equivalence of Definitions of Minimal Polynomial
 * Algebraic Element of Field Extension is Root of Unique Monic Irreducible Polynomial