Definition: GF(q)[x] = the collection of all
polynomials
of arbitrary degree with coefficients
in the finite field GF(q).
Definition: A polynomial p(x) is irreducible in
GF(q) if p(x) cannot be factored into a product of lower-degree
polynomials in GF(q)[x].
Definition: An irreducible polynomial 
GF(q)[x] of degree m is said to be primitive if the smallest
positive integer n for which p(x) divides 
is
.
Theorem: The roots 
of an mth-degree
primitive polynomial GF(q)[x] have order 
.
Theorem implies that the roots are primitive
elements in GF 
.