$\mathbb{F}[x^2,y^2,xy]$ is the polynomials in two variables whose terms all have even degrees. Of course, this generating set $x^2,y^2,xy$ is not algebraically independent, but I need to show that no generating set can be. 

There is a hint to work with systems of parameters. I've shown that the extension $\mathbb{F}[x^2,y^2,xy] \subset \mathbb{F}[x,y]$ is finite (with secondary generators x and y), so if there existed an independent generating set it would be a system of parameters for $\mathbb{F}[x,y]$. I'm having trouble figuring out what to do with this, though, since there is nothing wrong with $\mathbb{F}[x,y]$ having a system of parameters.

EDIT: All systems of parameters have the same cardinality. For $\mathbb{F}[x,y]$, this is cardinality 2 (for example, {x,y} is a system of parameters. So the problem reduces to showing that a set of two elements cannot generate $\mathbb{F}[x^2,y^2,xy]$. This seams reasonable enough, but I'm still not sure how to show this. Is it true in general that all minimal generating sets of an algebra have the same cardinality?