Definition:Minimal Polynomial

Definition
Let $K$ and $L$ be fields.

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

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

Let $K \left[{x}\right]$ be the ring of polynomial functions over $K$.

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

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

Also see

 * Minimal Polynomial is Unique
 * Minimal Polynomial is Irreducible