Primitive Element Theorem - Terminology

Terminology

Let be an arbitrary field extension. An element is said to be a primitive element for when

In this situation, the extension is referred to as a simple extension. Then every element x of E can be written in the form

where

for all i, and is fixed. That is, if is separable of degree n, there exists such that the set

is a basis for E as a vector space over F.

For instance, the extensions and are simple extensions with primitive elements and x, respectively ( denotes the field of rational functions in the indeterminate x over ).