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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Consider a simple extension $K(f)/K$ with $f\in K(x,y) \setminus K$.
This is a simple and elementary proof, I think.
Assume that $K(x,y)=K(t)$ for some $t\in K(x,y)$.