Transcendental extension that is not simple

Question

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?

Answer 1

Consider a simple extension $K(f)/K$ with $f\in K(x,y) \setminus K$.
Since $K$ is algebraically closed in $K(x,y)$, $f$ is transcedental over $K$ so that $\operatorname{trdeg}_K K(f)=1$.
However $\operatorname{trdeg}_K K(x,y)=2$ so that necessarily $K(f)\subsetneq K(x,y)$ and thus $K(x,y)/K$ is not simple.