Abstract
Abstract The main result of this chapter is that if F is a field and G is a finite subgroup of the multiplicative group of non-zero elements of F then G is cyclic. The proof is an elegant combination of elementary group theory and the theory of polynomial equations over a field. An immediate consequence of the main theorem is that if p is a prime number then the group of non-zero elements of ‘11../p”ll. under multiplication is cyclic. We shall also prove some general results about finite cyclic groups. For instance, if G is a cyclic group with n elements then G has precisely one subgroup of order d for each positive divisor d of n.
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