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

Also see

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