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Results tagged with nt.number-theory
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user 6776
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
14
votes
2
answers
1k
views
Subfields of a function field
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 …
- 601
6
votes
2
answers
2k
views
Computing the fixed field of an automorphism of a function field
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( …
- 601
0
votes
1
answer
342
views
Necessary/Sufficient condition/Algorithm that tells me a function field is a kummer extension
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 …
- 601
0
votes
0
answers
263
views
Computing the function field of a curve given as a subvariety of the Jacobian of its cover o...
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 …
- 601
0
votes
1
answer
406
views
The image of generator under an automorphism of a cyclic function field
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 …
- 601
0
votes
Accepted
The image of generator under an automorphism of a cyclic function field
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)^ …
- 601
0
votes
Accepted
Computing the fixed field of an automorphism of a function field
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 …
- 601