Let $\mathbb{F}$ be a field, why is $\mathbb{F}(x,y)$ not an algebric extension of $\mathbb{F}$? (Is $\mathbb{F}(x)$ an algebraic extension?)
*This is listed in my Field theory lecture notes as an important example for a finitely generated extension that is not algebraic and I don't know why it is not algebraic and if adding only one variable, $x$, is sufficient.