minimal polynomial

Let K/F be a field extension and κK be algebraic over F. The minimal polynomial for κ over F is a monic polynomial m(x)F[x] such that m(κ)=0 and, for any other polynomialPlanetmathPlanetmath f(x)F[x] with f(κ)=0, m divides f. Note that, for any element κ that is algebraic over F, a minimal polynomial exists (; moreover, because of the monic condition, it exists uniquely.

Given κK, a polynomial m is the minimal polynomial of κ if and only if m(κ)=0 and m is both monic and irreducible (

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