Minimal Polynomial (field Theory)

Minimal Polynomial (field Theory)

In field theory, a minimal polynomial is defined relative to a field extension E/F and an element of the extension field E. The minimal polynomial of an element, if it exists, is a member of F, the ring of polynomials in the variable x with coefficients in F. Given an element α of E, let J_α be the set of all polynomials f(x) in F such that f(α) = 0. The element α is called a root or zero of each polynomial in J_α. The set J_α is so named because it is an ideal of F. The zero polynomial, whose every coefficient is 0, is in every J_α since 0αi = 0 for all α and i. This makes the zero polynomial useless for classifying different values of α into types, so it is excepted. If there are any non-zero polynomials in J_α, then α is called an algebraic element over F, and there exists a monic polynomial of least degree in J_α. This is the minimal polynomial of α with respect to E/F. It is unique and irreducible over F. If the zero polynomial is the only member of J_α, then α is called a transcendental element over F and has no minimal polynomial with respect to E/F.

Minimal polynomials are useful for constructing and analyzing field extensions. When α is algebraic with minimal polynomial a(x), the smallest field that contains both F and α is isomorphic to the quotient ring F/<a(x)>, where <a(x)> is the ideal of F generated by a(x). Minimal polynomials are also used to define conjugate elements.