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Bill Allombert on Thu, 01 Dec 2022 12:55:42 +0100
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Re: The relationship between isometry group and Cyclotomic extensions of Galois group.
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- To: pari-users@pari.math.u-bordeaux.fr
- Subject: Re: The relationship between isometry group and Cyclotomic extensions of Galois group.
- From: Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>
- Date: Thu, 1 Dec 2022 12:54:36 +0100
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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