Remark

By the theorem, there are always primitive elements in finite fields.

If g is a primitive element of the finite field F, then the elements can be easily enumerated by their exponents with respect to g:

F = {0, g⁰ = 1, g¹ = g, g², g³, ..., g^{q - 2}}.

When written in this form, multiplication is given by modular arithmetic (modulus q - 1, putting 0 aside). This is very efficient, but addition is less convenient. Thus, we have the opposite to the usual form, where addition is a minor effort, but multiplication is harder.