Existence of Primitive Roots
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Existence of Primitive Roots
Recall from Section 2.1.2 that the order of an element in a finite group is the smallest such that . In this section, we prove that is cyclic by using the results of Section 2.5.1 to produce an element of of order for each prime power divisor of , and then we multiply these together to obtain an element of order . We will use the following lemma to assemble elements of each order dividing to produce an element of order . Lemma 2.5 Suppose have orders and , respectively, and that . Then has order . Proof. This is a general fact about commuting elements of any group; our proof only…
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