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 $m_{\alpha,K}(\alpha)=0$.

The minimal polynomial packages the algebraic relations of $\alpha$ over the base field and determines the simple extension $K(\alpha)$ up to $K$-isomorphism. It is defined inside the polynomial ring $K[x]$ and generates the kernel of the evaluation map $K[x]\to L$, $f\mapsto f(\alpha)$.

Examples:

• Over $\mathbb{Q}$, the minimal polynomial of $\sqrt{2}$ 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 $x-\alpha$.