Re: The relationship between isometry group and Cyclotomic extensions of Galois group.

[Hongyi Zhao <hongyi.zhao@gmail.com>]

On Thu, Dec 1, 2022 at 8:59 AM Hongyi Zhao <hongyi.zhao@gmail.com> wrote:
>
> Hi here,
>
> It seems that there are subtle relationship between isometry group [1]
> and Cyclotomic extensions of Galois group [2]. But I can't give a
> clean description of this relationship. Any tips will be appreciated.

In fact, I'm considering the relationship between the Crystallographic
point group [1], which is a type of isometry group, and the
Cyclotomic extensions of Galois group, which, if I understand correct,
can be represented as Cyclotomic matrix group [2].

So, to be more precise, my question is: what's the relationship
between Crystallographic point group and Cyclotomic matrix group?

[1] https://en.wikipedia.org/wiki/Crystallographic_point_group
[2] Cyclotomic matrix group is a group whose matrices all have
cyclotomic entries, as described here [3].
[3] https://docs.gap-system.org/doc/ref/chap44.html#X850821F78558C829

Best regards,
Zhao

> [1] https://en.wikipedia.org/wiki/Isometry_group
> [2] https://en.wikipedia.org/wiki/Galois_group#Cyclotomic_extensions
>
> Regards,
> Zhao
> --
> Assoc. Prof. Hongsheng Zhao <hongyi.zhao@gmail.com>
> Theory and Simulation of Materials
> Hebei Vocational University of Technology and Engineering
> No. 473, Quannan West Street, Xindu District, Xingtai, Hebei province