Primitive element (finite field)


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.

Properties

Number of primitive elements

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.