Question

Prove that for any positive integer n a field f can have at most a finite number of elements of multiplicative order at most n

Answer 1

Let's assume multiplicative order is infinite. Then $x^k=1, \forall k=1(1)n$. In the field $F$ the solution of the polynomial $x^k-1=0$ can have at most $k$ distinct solutions. Hence for any $k=1(1)n$ we cannot have infinite roots. And thus the result follows.