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

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Thu, Dec 01, 2022 at 08:59:30AM +0800, Hongyi Zhao 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.
> 
> [1] https://en.wikipedia.org/wiki/Isometry_group
> [2] https://en.wikipedia.org/wiki/Galois_group#Cyclotomic_extensions

You are misreading. 
Cyclotomic extensions are a special kind of Galois extensions of fields,
not extensions of Galois groups.

I suggest you should study the theory of representation of groups, which
allow to represent groups as group of unitary matrices with coefficients
in a cyclotomic field.

Cheers,
Bill