Talk:Profinite integer

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Someone should add: the abelianization map identifies the absolue Galois group G of Q with ${\displaystyle {\widehat {\mathbb {Z} }}^{*}}$ (class field theory?) Put in another way the non-abelian-ness of profinite integers are hidden in the commutator subgroup of G (this stuff is beyond me). -- Taku (talk) 02:55, 27 April 2015 (UTC)Reply

I have a problem with the relation

${\displaystyle {\hat {\mathbb {Z} }}\subset \prod _{n}\mathbb {Z} /n\mathbb {Z} ,}$

because on the left side there is an uncountable set and on the right side there is a countable product of finite sets.

The mentioned problem does not exist with

${\displaystyle {\widehat {\mathbb {Z} }}=\prod _{p}\mathbb {Z} _{p},}$

because the ${\displaystyle \mathbb {Z} _{p}}$ are (as complete sets) already uncountable, and ${\displaystyle {\widehat {\mathbb {Z} }}=\varprojlim \mathbb {Z} /n\mathbb {Z} }$ is, of course, OK . –Nomen4Omen (talk) 07:04, 30 May 2021 (UTC)Reply

Solved! The right side is an infinite direct product. –Nomen4Omen (talk) 19:19, 30 May 2021 (UTC)Reply

I think this statement in the last section is wrong:

${\displaystyle {\begin{aligned}\mathbb {A} _{\mathbb {Q} }^{\times }/\mathbb {Q} ^{\times }&\cong (\mathbb {R} \times {\hat {\mathbb {Z} }})/\mathbb {Z} \\&={\underset {\leftarrow }{\lim }}\mathbb {(} {\mathbb {R} }/m\mathbb {Z} )\\&={\underset {x\mapsto x^{m}}{\lim }}S^{1}\\&={\hat {\mathbb {Z} }}\end{aligned}}}$

The abelian Galois group of ${\displaystyle \mathbb {Q} }$ is ${\displaystyle {\hat {\mathbb {Z} }}^{\times }}$ and not ${\displaystyle {\hat {\mathbb {Z} }}}$. Furthermore, this is an isomorphism of abstract groups, but not an isomorphism of topological groups.

93.132.116.137 (talk) 20:13, 2 July 2021 (UTC)Reply