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This question is inspired by Pace Nielsen's recent question Does a left basis imply a right basis, without AC?. For any field $k$, the field $k(x)$ of rational functions in one variable has an ...

Is there a field which is the union of finitely many proper subfields?

Consider $\mathbf{Q}(x,y)$, the rational functions in $x$ and $y$, as a vector space over $\mathbf{Q}$. Let $\sigma$ be the map interchanging $x$ and $y$. Is there a basis for $\mathbf{Q}(x,y)$ ...