Suppose that $K$ is an algebraically closed field. There is a statement:
If $K$ is not the algebraic closure of a finite field, then $K^*$ contains free abelian groups of arbitrarily large finite rank.
Is it true? And why?
Moreover, is
$K$ is not the algebraic closure of a finite field, if and only if $K^*$ contains free abelian groups of arbitrarily large finite rank.
True?
Thanks very much.