Re: Artin's method for class field computation

Hi Lucas,

There would be no problem implementing the method you suggest to deal 
with prime power degree extensions, but we chose to implement 
bnrclassfield by relying on the historical function rnfkummer, and at 
the moment we don't have plans to implement the other method although it 
could be useful in some cases.
Algorithmically, the best method depends on the ramification of the 
intermediate extensions involved. What is costly in explicit class field 
theory using kummer theory is that to construct a cyclic extension of 
degree n of K, you have to compute the class group and units of K(zeta_n).
- If you directly do the extension of degree p^m, you have to do this 
computation for K(zeta_p^m).
- If you use a tower of degree p extensions K_m/K_{m-1}/.../K_1/K, then 
the most costly step is the last one where you have to do the 
computation for K_{m-1}(zeta_p).
So it depends on which field is harder between K(zeta_p^m) and 
K_{m-1}(zeta_p), and this depends on the situation.

In terms of references, I think what you call the Artin method is 
described in the paper of Claus Fieker "Computing class fields via the 
Artin map". In our implementation, we never use the complete Artin map, 
but we use the splitting type of primes, which contains less information 
than the full Artin map but enough to characterise the extension.

Aurel

On 30/04/2021 11:16, lucas legrand wrote:
> Hi,
>
> Looking at 'kummer.c', I (think I) understand that your current 
> bnrclassfield implementation break the looked for extension into 
> intermediate prime degree extensions and then apply rnfkummer to each 
> of them, following closely Hecke's method in Cohen Vol. 2.
>
> Maybe I'm mistaken, but I can't find anything in sources related to 
> Artin's method which uses Artin reciprocity map and can deal with 
> prime power degree extensions at once. Is there reasons Artin's method 
> is not implemented ?  Maybe it's unpublished, or you have plan for this ?
>
> Well this question is just for curiosity, since I'm actually playing 
> around with Artin's method for understanding purposes.
>
> Lucas