In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF(q) is called a primitive element if it is a [[primitive root of unity|primitive (q − 1) th root of unity]] in GF(q) GF(q) can be written as α for some natural number i . If q is a prime number, the elements of GF(q) can be identified with the integers modulo q. In this case, a primitive element is also called a primitive root modulo q. For example, 2 is a primitive element of the field GF(3) and GF(5) , but not of GF(7) since it generates the cyclic subgroup {2, 4, 1} of order 3; however, 3 is a primitive element of GF(7) . 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 GF(q) is φ(q − 1) , where φ is Euler's totient function, which counts the number of elements less than or equal to m that are coprime to m . This can be proved by using the theorem that the multiplicative group of a finite field GF(q) is cyclic of order q − 1 , and the fact that a finite cyclic group of order m contains φ(m) generators.