Definition:Minimal Polynomial

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

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

Definition 1
The minimal polynomial of $\alpha$ over $K$ is the unique monic polynomial $f \in K[x]$ of smallest degree such that $f(\alpha) = 0$.

Definition 2
The minimal polynomial of $\alpha$ over $K$ is the unique irreducible, monic polynomial $f \in K \left[{x}\right]$ such that $\mu \left({\alpha}\right) = 0$.

Definition 3
The minimal polynomial of $\alpha$ over $K$ is the unique monic polynomial $f$ that generates the kernel of the evaluation homomorphism $K[x] \to L$ at $\alpha$.

Note that $\mu$ depends only on $\alpha$ and $K$.

Also see

 * Minimal Polynomial is Unique
 * Minimal Polynomial is Irreducible