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 (http://planetmath.org/ExistenceOfTheMinimalPolynomial); 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 (http://planetmath.org/IrreduciblePolynomial).

Title	minimal polynomial
Canonical name	MinimalPolynomial
Date of creation	2013-03-22 13:20:11
Last modified on	2013-03-22 13:20:11
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	13
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 11C08
Classification	msc 11R04
Classification	msc 12F05
Classification	msc 12E05
Related topic	DegreeOfAnAlgebraicNumber