Let $K$ be a field of characteristic $\neq 2$. Let $C_4=\langle\sigma\rangle$ act on $K^2$ by $\sigma (a,b)=(-b,a)$. This translates to $\sigma:s\mapsto t, t \mapsto -s$ on $K(s,t)$, and it is "easily seen" that $K(s,t)^{C_4}=K(u,v)$ for
$$u=\frac{s^2-t^2}{st}$$
and
$$v=s^2+t^2$$
Another example is $K(s,t)^{C_3}=K(u,v)$, where $C_3=\langle\sigma\rangle$ acts on $K^2$ by $\sigma(a,b)=(-b,a-b)$, so $\sigma:s\mapsto t,t\mapsto -s-t$, and so
$$u=\frac{s^2+t^2+st}{st(s+t)}$$
and
$$v=\frac{s^3-3st^2-t^3}{st(s+t)}$$
I understand why these are homogeneous invariants, but why do they necessarily form a minimal basis of $K(s,t)^G$?