# Minimal polynomial

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.

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

*geometrical multiplicity* of λ and is the size of the largest

Jordan block corresponding to λ.

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In

field theory, a

**minimal polynomial** is a

polynomial *m*(

*x*) in the field

**Z**_{p} (with

*p* prime), such that, if we have the field

**F**=

**Z**_{p}(α), it is the polynomial of least

degree with

*m*(α)=0.

The minimal polynomial is unique, and if we have some irreducible polynomial *f*(*x*) with *f*(α)=0, then *f* is the minimal polynomial of α.