TITLE: A field $F$ has at most a finite number of elements of order $\leq$ $n$ for any $n$ in integers.
QUESTION [7 upvotes]: A field $F$ has at most a finite number of elements of order $\leq n$ for any $n$ in integers. 
How can I prove this? I thought it's related to the fact that a polynomial of degree $n$ would have at most $n$ roots over $F$. But I am not sure. Would appreciate hints.

REPLY [5 votes]: Hint: by pigeonholing, if there are infinitely many elements of order $\le n$ then for some $k\le n$ there are infinitely many elements of order $k,\:$ hence $x^k - 1$ has infinitely many roots $\:\Rightarrow\Leftarrow$