Transcendental extension that is not Simple

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?

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Suppose to the contrary that it is simple and generated by some $t$. Write $x$ and $y$ as rational functions in $t$. Then show there is some relation between them, which contradicts independence. –  Qiaochu Yuan Jun 15 '12 at 13:41

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.