Relatively prime integer

Question

Please help me to answer the following problem:

Let $F$ a field.

Show that if $m$ and $n$ are relatively prime integer, and $r,s\in F^{\times}$ satisfy $r^m=s^n$ then there are $u,v\in F^{\times}$ such that $r=u^n$ and $s=v^m$.

Thanks

Answer 1

Find $a,b\in\Bbb Z$ with $an+bm=1$, which is possible because $n$ and $m$ are coprime. Then, $r^{1-an}=r^{bm}=s^{bn}$ so we get $r = s^{bn}r^{an} = (s^br^a)^n$. Similarly, $s^{1-bm}=s^{an}=r^{am}$ so $s=r^{am}s^{bm}=(r^as^b)^m$. Hence, you can pick $u=s^br^a$ and $v=r^as^b$.