system of intermediate fields of extension is infinite

Question

I need help to prove that system of intermediate fields of extension $\mathbb{k}(x,y)$, of field $\mathbb{k}(x^p,y^p) \ char(\mathbb{k}) = p > 0$ is infinite

Answer 1

For $m\ne n$ if $$k(x^p,y^p,x^{mp+1} +y)= k(x^p,y^p,x^{np+1} +y)$$ then $$k(x^p,y^p,x^{mp+1} +y,x^{np+1} +y)=k(x,y)$$

has degree $p$ over $k(x^p,y^p)$ which is a contradiction.