Definition:Minimal Polynomial/Definition 3

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 $f \in K[x]$ that generates the kernel of the evaluation homomorphism $K \left[{x}\right] \to L$ at $\alpha$.

That is, such that for all $g \in K[x]$:
 * $g(\alpha)=0$ $f$ divides $g$.

Also see

 * Equivalence of Definitions of Minimal Polynomial
 * Nonzero Ideal of Polynomial Ring over Field has Unique Monic Generator