In field theory, a primitive element of a finite field is a generator of the multiplicative group of the field. In other words, is called a primitive element if it is a primitive th root of unity in ; this means that each non-zero element of can be written as for some natural number .
If is a prime number, the elements of can be identified with the integers modulo . In this case, a primitive element is also called a primitive root modulo .
For example, 2 is a primitive element of the field and, but not of since it generates the cyclic subgroup of order 3; however, 3 is a primitive element of . The minimal polynomial of a primitive element is a primitive polynomial.
The number of primitive elements in a finite field is, where is Euler's totient function, which counts the number of elements less than or equal to that are coprime to . This can be proved by using the theorem that the multiplicative group of a finite field is cyclic of order, and the fact that a finite cyclic group of order contains generators.