Originally Posted by

**Gusbob** Since $\displaystyle X$ and $\displaystyle Y$ are variables, surely you can just replace $\displaystyle X$ and $\displaystyle Y$ by two linearly independent variables which may or may not depend on $\displaystyle X$ and $\displaystyle Y$, say $\displaystyle f(X,Y),g(X,Y)$. Then $\displaystyle X\mapsto f(X,Y),Y\mapsto g(X,Y),1\mapsto 1$ gives a ring isomorphism between $\displaystyle \mathbb{Q}[X,Y]$ and $\displaystyle \mathbb{Q}[f(X,Y),g(X,Y)]$.

EDIT: I realised the restriction of linearly independent is not enough (see $\displaystyle X$ and $\displaystyle X^2$). But if you specify that $\displaystyle f(X,Y)^i$ and $\displaystyle g(X,Y)^j$ are linearly independent for all choices of $\displaystyle i,j\in \mathbb{N}$, it should be enough.