# Function field of the projective line

Suppose I chose two rational functions, say, $$u = \frac{t(4+t)^5}{(1+4t)^5}, \qquad v = \frac{t^5 (4+t)}{(1+4t)}.$$ Then I know that $K(X):= \mathbf{C}(u,v)$ is the function field of the projective line (Proof: If $K(Y):=\mathbf{C}(t)$, then there is an inclusion $K(Y) \subseteq K(X)$, and hence a surjection $Y \simeq \mathbf{P}^1 \rightarrow X$, and so $X$ must have genus $0$.) From this follows that $K(X) \simeq \mathbf{C}(s)$ for some $s \in \mathbf{C}(t)$. Is there a practical easy algorithm to explicitly construct such an $s$, and, moreover, write $s$ as a rational function of $X$ and $Y$? Is there an easy way at least to determine the degree of the map $Y \rightarrow X$?

In case you were wondering, the specific choice of $X$ and $Y$ was motivated by the question:

http://mathoverflow.net/questions/50804/deciding-whether-a-given-power-series-is-modular-or-not

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Isn't the inclusion the other way around: $\mathbf{C}(u,v)\subseteq \mathbf{C}(t)$? The rest of your agrument seems to check out. Can't help with your question, but can you find anything helpful by searching for `Lüroth's theorem'? –  Jyrki Lahtonen Jun 15 '11 at 11:26