Hi,

By GL_{n}(F) I assume you mean the group of non-singular n by n matrices with entries from F. Let x in F be non-zero and A in GL_{n}(F) the diagonal matrix with A(1,1)=x and all other entries on the diagonal 1. This establishes a one to one correspondence between non-zero elements of F and a subset of GL_{n}(F). Clearly then F is finite provided GL_{n}(F) is finite.