# Minimal polynomial

In mathematics, the **minimal polynomial** of an object α is the monic polynomial *p* of least degree such that *p*(α)=0. The properties of the minimal polynomial depend on the algebraic structure to which α belongs.

## Field theory

In field theory, given a field extension *E*/*F* and an element α of *E* which is algebraic over *F*, the **minimal polynomial** of α is the monic polynomial *p*, with coefficients in *F*, of least degree such that *p*(α) = 0. The minimal polynomial is irreducible, and any other non-zero polynomial *f* with *f*(α) = 0 is a multiple of *p*.

## Linear algebra

In linear algebra, the **minimal polynomial** of an *n*-by-*n* matrix *A* over a field **F** is the monic polynomial *p*(*x*) over **F** of least degree such that *p*(*A*)=0. Any other polynomial *q* with *q*(*A*) = 0 is a (polynomial) multiple of *p*.

The following three statements are equivalent:

- λ∈
**F**is a root of*p*(*x*), - λ is a root of the characteristic polynomial of
*A*, - λ is an eigenvalue of
*A*.

The multiplicity of a root λ of *p*(*x*) is the *geometric multiplicity* of λ and is the size of the largest Jordan block corresponding to λ and the dimension of the corresponding eigenspace.

The minimal polynomial is not always the same as the characteristic polynomial. Consider the matrix , which has characteristic polynomial . However, the minimal polynomial is , since as desired, so they are different for . That the minimal polynomial always divides the characteristic polynomial is a consequence of the Cayley–Hamilton theorem.