For field $F$, $F^n \simeq F^m$ iff $m=n$

Question

I was reading this question.

I understand Alex's answer using tensor products except the last part. I guess his conclusion is if $F^n \simeq F^m$ have the same module structure then $n=m$, but I don't know why this is true. Can anyone elaborate on this?

Answer 1

Note that a module over a field is a vector space. So, Alex's answer is referring to the dimension theorem for vector spaces.