Definition:Minimal Polynomial

Definition
Let $K \subset L$ be fields and $\alpha \in L$ be algebraic over $K$.

Let $K \left[{x}\right]$ be the Ring of Polynomial Functions over $K$.

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

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

Also see

 * Minimal Polynomial is Unique
 * Minimal Polynomial is Irreducible