For the construction we are going to discuss on the
next page,
we need a prime p and a natural number x such that
the powers of x modulo p run through the entire set {1,
..., p - 1}.
An element g from Z/pZ is called a primitive element of Z/pZ if every element of Z/pZ\{0} is a power of g.
For every prime p there exist primitive elements; but we cannot say a priori which ones.
For each prime p there exists a primitive element in Z/pZ.
For the proof we need a little bit more theory than we have developed so far. It will be postponed to Chapter 7. But for explicitly given primes p such a primitive element can easily be found by trial and error.