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

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

On Thu, Dec 1, 2022 at 7:54 PM Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> wrote:
>
> 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.

Thank you for your response and explanation. Relatively speaking, I
understand more about group representation theory than Galois groups,
which makes led me to this inappropriate question.

> 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.

I also know the important fact that every matrix representation of a
finite group G is conjugate, a.k.a., equivalent, to a unitary
representation [1]. But in fact, it is sometimes very difficult to
find a nice unitary representation or even find such a unitary
representation.

Anyway, let me return to my real problem discussed here. In fact, the
background of my question is as follows:

In fact, I'm considering the relationship between the Crystallographic
point group [2], which is a type of isometry group, and the Cyclotomic
matrix group which is a matrix group whose elements all have
cyclotomic entries, as described here [3].

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

[1] http://math.uchicago.edu/~may/REU2018/REUPapers/Kim.pdf, [Theorem
3.8 (p. 6)]
[2] https://en.wikipedia.org/wiki/Crystallographic_point_group
[3] https://docs.gap-system.org/doc/ref/chap44.html#X850821F78558C829

> Cheers,
> Bill

Best,
Zhao