Let $K/F$ be a field extension and $\kappa\in K$ be algebraic over $F$. The minimal polynomial for $\kappa$ over $F$ is a monic polynomial $m(x)\in F[x]$ such that $m(\kappa)=0$ and, for any other polynomial $f(x)\in F[x]$ with $f(\kappa)=0$, $m$ divides $f$. Note that, for any element $\kappa$ that is algebraic over $F$, a minimal polynomial exists; moreover, because of the monic condition, it exists uniquely.

Given $\kappa\in K$, a polynomial $m$ is the minimal polynomial of $\kappa$ if and only if $m(\kappa)=0$ and $m$ is both monic and irreducible.