simple extension

A simple extension is the most economical way to grow a field: you add exactly one new element and take the smallest field containing it. Everything in the new field is then built from that single generator using the four arithmetic operations and the base field. It is the field-theoretic analogue of a cyclic group, where one element generates the whole structure, and it is the basic building block from which larger extensions are assembled.

Precisely, an extension L of K is simple if L = K(a) for some single element a, called a primitive element. There are two flavors. If a is algebraic over K with minimal polynomial of degree n, then K(a) is isomorphic to K[x]/(minimal poly) and has degree n with basis {1, a, ..., a^(n-1)}. If a is transcendental, then K(a) is isomorphic to the rational function field K(x), of infinite degree.

Simple extensions are pervasive because of the primitive element theorem: every finite separable extension is in fact simple, so over fields of characteristic 0 or finite fields, working with a single generator and one minimal polynomial loses no generality. Not every extension is simple, though — inseparable extensions can require two or more generators.