Suppose you have a a curve $C$ defined by an equation in $x$ and $y$. There is a map from $C$ to $\mathbb{P}_1$ by projection onto $x$. This corresponds to a separable extension of function fields with primitive elements the function $y$. Lets call this field extension $F\to K$. Now take a quotient curve of $C$ by a subgroup of the automorphisms of $C$. Its function field, H, fits in the diagram $F \to H \to K$. Are there any techniques for calculating a primitive element for the extension $H \to K$ in terms of other information given.
Some suggestions I got where applying the norm or trace map to the primitive element above but this does not work.