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Someone should add: the abelianization map identifies the absolue Galois group G of Q with
Z
^
∗
{\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 ]
Z
^
⊂
∏
n
Z
/
n
Z
,
{\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
Z
^
=
∏
p
Z
p
,
{\displaystyle {\widehat {\mathbb {Z} }}=\prod _{p}\mathbb {Z} _{p},}
because the
Z
p
{\displaystyle \mathbb {Z} _{p}}
complete sets) already uncountable, and
Z
^
=
lim
←
Z
/
n
Z
{\displaystyle {\widehat {\mathbb {Z} }}=\varprojlim \mathbb {Z} /n\mathbb {Z} }
Nomen4Omen (talk ) 07:04, 30 May 2021 (UTC) [ reply ]
