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Colin Weir, suggested the following algorithm to solve the problem in non-rational case, I thought for the sake of others who probably have the same question, I'll post it, here: Suppose that $\sigma …

Let say we have a function field $k(x,y)$ defined by $f(x,y)$ over $k$, with $\sigma \in Aut(k(x,y)/k)$ and. Suppose, I'm not that out of luck, so that either of $\prod \sigma^i(x)$ or $\sum \sigma^i( …

I'm reading the proof of Lemma 4.1 [1] which says: "Let $F = K(x,y), y^q = f(x)$, where $q$ is a prime different from characteristic of $K$. Let $Z := Gal(F/K(x))$ and we have $Z < G < Aut(F/K)$ Then …

OK, finally, I think I got it, but it is not that trivial to simply be omitted from the proof (If I complicated it and there's is a straight forward way to see it please tell me): We have $\sigma(y)^ …

I read following paragraph from: G. Tamme, Teilkörper höheren Geschlechts eines algebraischen Funktionenkörpers, Arch. Math. 23 (1972), 257--259 Here $C$ is a curve of genus $\ge 2$ and $J$ is the …

I start my question with an example. Suppose $F/K$ be the function field generated by $x^n - yx^{n-1} - 1 = 0$. It is not a cyclic over K(y), but if I set $t = yx^{n-1}$ then we have $K(x,t) \subset K …

Is there an algorithm for generating (some or all) subfields of a certain genus of a given function field (even a random one,I mean for example generating a random elliptic subfield of a certain given …