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,
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 of order 3 {2,4,1}; however, 3 is a primitive element of GF(7). The minimal polynomial of a primitive element is a primitive polynomial.
Number of primitive elements
The number of primitive elements in a finite field GF(q) is φ(q - 1), where φ(m) is Euler's totient function, which counts the number of elements less than or equal to m which are relatively prime 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.