The Group of Units
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We saw in Chapter 5 that for each n, the set Un of units in ℤn forms a group under multiplication. Our aim in this chapter is to understand more about multiplication and division in ℤn by studying the structure of this group. An important result is that if n = pe, where p is an odd prime, then Un is cyclic; following a commonly-used strategy, we shall prove this first for n = p, and then deduce it for n = pe. As often happens in number theory, the prime 2 is exceptional: although U2 and U4 are cyclic, we shall see that the group U2e is not cyclic for e ≥ 3, although in a certain sense it is nearly cyclic. Using the Chinese Remainder Theorem, we can use our knowledge of the prime-power case to deduce the structure of Un for arbitrary n. As an application, we will continue the study of Carmichael numbers, begun in Chapter 4.
KeywordsCyclic Group Primitive Root Chinese Remainder Theorem Congruence Class Finite Abelian Group
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