information:extraction
Minimal polynomial

If θ is an element of an associative K -algebra L, the polynomial evaluation at θ is the unique algebra homomorphism φ from K into L that maps X to θ and does not affect the elements of K itself (it is the identity map on K). It consists of substituting X with θ in every polynomial. That is,

The image of this evaluation homomorphism is the subalgebra generated by θ, which is necessarily commutative.If φ is injective, the subalgebra generated by θ is isomorphic to K. In this case, this subalgebra is often denoted by K. The notation ambiguity is generally harmless, because of the isomorphism.

If the evaluation homomorphism is not injective, this means that its kernel is a nonzero ideal, consisting of all polynomials that become zero when X is substituted with θ. This ideal consists of all multiples of some monic polynomial, that is called the minimal polynomial of θ. The term minimal is motivated by the fact that its degree is minimal among the degrees of the elements of the ideal.

There are two main cases where minimal polynomials are considered.

In field theory and number theory, an element θ of an extension field L of K is algebraic over K if it is a root of some polynomial with coefficients in K. The minimal polynomial over K of θ is thus the monic polynomial of minimal degree that has θ as a root. Because L is a field, this minimal polynomial is necessarily irreducible over K. For example, the minimal polynomial (over the reals as well as over the rationals) of the complex number i is X 2 + 1 {\displaystyle X^{2}+1}. The cyclotomic polynomials are the minimal polynomials of the roots of unity.

In linear algebra, the n × n square matrices over K form an associative K -algebra of finite dimension (as a vector space). Therefore the evaluation homomorphism cannot be injective, and every matrix has a minimal polynomial (not necessarily irreducible). By Cayley–Hamilton theorem, the evaluation homomorphism maps to zero the characteristic polynomial of a matrix. It follows that the minimal polynomial divides the characteristic polynomial, and therefore that the degree of the minimal polynomial is at most n.