Test whether a finite field element is a primitive element of its ambient field:

Test elements of a finite field in polynomial representation:

Test elements of a finite field in exponential representation:

Arguments that are not FiniteFieldElement objects yield False:

Implement a discrete logarithm function for a given base element:

The base element needs to be primitive:

Construct a field with exponential element representation using the minimal polynomial of b:

Construct a field isomorphism that maps b to the field generator of :

Define the discrete logarithm function:

Compute the discrete logarithm for a nonzero element of ℱ:

a is a primitive element of its ambient field if every nonzero element of the field is an integer power of a:

a is a primitive element of if it is a generator of and its minimal polynomial is primitive:

Use MinimalPolynomial to find the minimal polynomial of a:

This shows that a is a generator of :

Use PrimitivePolynomialQ to show that f is primitive:

The multiplicative order of a primitive element is one less than its ambient field size:

Use MultiplicativeOrder to compute the multiplicative order of a:

A finite field with q elements contains EulerPhi[q-1] primitive elements: