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 (q − 1)th root of unity in GF(q); this means that each non-zero element of GF(q) can be written as αi 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.