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This might be a dumb question, but I just realized that the field of rational functions $k(x)$ has a subfield consisting of rational functions which can be expressed as fractions of two even polynomials. It came to me when I was thinking about Veronese embeddings. I have never encountered such a field(or, more precisely, such a description) before. If I am not wrong, it is the unique degree 2 subfield over $k$ of $k(x)$. How did I not know such a relation! I wonder if there are any interesting topics regarding this(or higher dimensional analogue, or higher degree analogue).

Gyujin Oh
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1 Answers1

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Lüroth's Theorem:

Let $k$ be a field and $L$ be an intermediate field between $k$ and $k(x)$, for some indeterminate $x$. Then there exists a rational function $f(x) \in k(x)$ such that $L=k(f(x))$. In other words, every intermediate extension between $k$ and $k(x)$ is simple.

So now in your question, each $k(x^n)$ is a subfield of $k(x)$.

Fei Hu
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