Extension Field

Related Concepts

Field; Irreducible Polynomial; Number Field; Primitive Element; Optimal Extension Fields (OEFs)

Definition

An extension field is a field with certain mathematical structure constructed from another field and one or more roots of polynomials over that field.

Theory

Let $F = (S,+,\times )$ be a field and let f(x) be a monic irreducible polynomial of degree d over F. That is, let f(x) be a polynomial

$$ f(x) = {x}^{d} + {f}_{ d-1}{x}^{d-1} + \cdots + {f}_{ 1}x + {f}_{0}. $$

Let α denote a root of this polynomial. Then the set of elements generated from field operations on elements of F and α is itself a field, denoted F(α). The field F(α) is called an extension field of F of extension degree d. F is a subfield of F(α) since F is a subset of F(α) and is itself a field.

The definition given here is for a simple extension field. In general, an extension field of a field F is any field that contains F as a subfield; an example of a non-simple extension of F is F(α,;β),...