Re: Artin's method for class field computation

[Aurel Page <aurel.page@normalesup.org>]

    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  Maybe it's unpublished, or you have plan for this ?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