let $C_3=\langle\sigma\rangle$ and let $\sigma$ act on $K(s,t)$ by $s \mapsto t$ and $t \mapsto -s-t$
I want to prove that $K(s,t)^{C_3}=K(u,v)$ with $u=\frac{s^2+t^2+st}{st(s+t)}$ and $v=\frac{s^3-3st^2-t^3}{st(s+t)}$ where $K(s,t)^{C_3}$ is the fixed field. $\supseteq$ can just be calculated, but i got problems showing the other direction. The best methode usually seems to be showing that $[K(s,t):K(u,v)]=3$, but i have no idea how i would do so in this case, since i cant see the minimal polynomial.
Is there maybe a better way to solve the problem in this case or can someone tell me how to find the minimal polynomial?
Thx in advance