Minimal polynomial over a field

Let $K\subseteq L$ be fields and let $\alpha\in L$ be algebraic over $K$ . The minimal polynomial of $\alpha$ over $K$ is the unique monic irreducible polynomial $m_{\alpha,K}(x)\in K[x]$ such that $m_{\alpha,K}(\alpha)=0$ .

Remarks

It generates the kernel of the evaluation map $K[x]\to L$ , $f\mapsto f(\alpha)$ , and determines the simple extension $K(\alpha)$ up to $K$ -isomorphism.

Examples

Over $\mathbb Q$ , the minimal polynomial of $\sqrt2$ is $x^2-2$ .
Over $\mathbb R$ , the minimal polynomial of $i$ is $x^2+1$ .
If $\alpha\in K$ , then the minimal polynomial is $x-\alpha$ .