minimal polynomials

Noun

minimal polynomial (plural minimal polynomials)

1. For a given square matrix M over some field K, the smallest-degree monic polynomial over K which, when applied to M, yields the zero matrix.
   • Each root of the minimal polynomial of a matrix M is an eigenvalue of M and a root of its characteristic polynomial. (A root of the minimal polynomial has a multiplicity that is less than or equal to the multiplicity of the same root in the characteristic polynomial. Thus the minimal polynomial...
2. (field theory) Given an algebraic element α of a given extension field of some field K, the monic polynomial of smallest degree of which α is a root.
   • […]we are given an element #92;alpha#92;inE, and want to compute the minimal polynomial #92;phi#92;inF#91;X#93; of #92;alpha over F.2005, Victor Shoup, A Computational Introduction to Number Theory and Algebra, Cambridge University Press, page 438