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.
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.
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