The German school thinks differently. There is a different (well known) algorithm due to Gauss. Take an arbitrary number a coprime to p. Find its order. If it is the primitive root, we are done. If not, choose another b and check whether order of ab is greater than the order of a. If it is, replace a by ab and repeat the procedure. If the order of ab is less than order of b, look for another b such that order of ab is greater than a. We get a sequence of elements whose orders are strictly increasing. Since the order is finite, the process must stop at some point yielding a primitive root. It is discussed in one of the books, either Zassenhaus and Pohst, or the slim volume written by Pohst.
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