## You are here

Homeminimal polynomial

## Primary tabs

# minimal polynomial

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.

Related:

DegreeOfAnAlgebraicNumber

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

11C08*no label found*11R04

*no label found*12F05

*no label found*12E05

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Oct 21

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag