A field extension L/K is called a simple extension if there exists an element θ in L with
The element θ is called a primitive element, or generating element, for the extension; we also say that L is generated over K by θ.
A primitive element of a finite field is a generator of the field's multiplicative group. When said at greater length: In the realm of finite fields, a stricter definition of primitive element is used. The multiplicative group of a finite field is cyclic, and an element is called a primitive element if and only if it is a generator for the multiplicative group. The distinction is that the earlier definition requires that every element of the field be a quotient of polynomials in the primitive element, but within the realm of finite fields the requirement is that every nonzero element be a pure power.
The only field contained in L which contains both K and θ is L itself. More concretely, this means that every element of L can be obtained from the elements of K and θ by finitely many field operations (addition, subtraction, multiplication and division).
In other words every element of K(θ) can be written as a quotient of two polynomials in θ with coefficients from K.
Given a field K the simple extensions K(θ) can be completely classified using the polynomial ring K[X] in one indeterminate,