Hongyi Zhao on Thu, 01 Dec 2022 13:22:45 +0100 |
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Re: The relationship between isometry group and Cyclotomic extensions of Galois group. |
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