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.