$C(X,Y)$ is a algebraic extension of $C(X^n+Y^n, XY)$

Question

How to prove that a rational function field $\mathbb{C}(X, Y)$ is algebraic extension of its subfield $\mathbb{C}(X^{n}+Y^{n},XY)$?

It should be ok if $X$ is algebraic over $\mathbb{C}(X^{n}+Y^{n}, XY)$, but I can’t prove that.

Answer 1

$X^n,Y^n$ are roots of $T^2-(X^n+Y^n)T+(XY)^n=0$. Therefore they are algebraic over$\mathbb{C}(X^{n}+Y^{n}, XY)$. And $X,Y$ are algebraic over $\mathbb{C}(X^n,Y^n)$.