TITLE: Transcendental extension that is not simple
QUESTION [6 upvotes]: Let $K$ be a field and $x, y$ be independent variables. How can I show that $K(x, y)/K$ is not a simple extension?

REPLY [2 votes]: This is a simple and elementary proof, I think.
Assume that $K(x,y)=K(t)$ for some $t\in K(x,y)$.
Since $x\in K(t)$, there exist (coprime and nonzero) polynomials $u(z),v(z)\in K[z]$ such that $x=\frac{u(t)}{v(t)}$.
Consider now the polynomial $f(z)=xv(z)-u(z)\in K(x)[z]$: clearly it is not $0$ and $f(t)=0$, which means that $t$ is algebraic over $K(x)$, that is, $K(x)[t]$ is an algebraic extension of $K(x)$. But then $K(x,y)=K(t)=K(x,t)=K(x)(t)=K(x)[t]$ is algebraic over $K(x)$, a contradiction.