Let be a field extension and be algebraic over . The minimal polynomial for over is a monic polynomial such that and, for any other polynomial with , divides . Note that, for any element that is algebraic over , a minimal polynomial exists; moreover, because of the monic condition, it exists uniquely.
Given , a polynomial is the minimal polynomial of if and only if and is both monic and irreducible.
![$ m(x)\in F[x]$](http://images.planetmath.org:8080/cache/objects/3850/l2h/img6.png "$ m(x)\in F[x]$"){kind=small}
![$ f(x) \in F[x]$](http://images.planetmath.org:8080/cache/objects/3850/l2h/img8.png "$ f(x) \in F[x]$"){kind=small}
