What is the automorphism group of the field of all constructible numbers?
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Let $Omegasubseteq mathbbC$ be the field of all constructible numbers (i.e. $Omega$ is the smallest subfield of $mathbbC$ which is closed under taking square roots). What is known about the automorphismgroup of the field $Omega$?
reference-request field-theory galois-theory profinite-groups
asked Apr 4 '15 at 11:36
user 59363
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up vote
9
down vote
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Let $Omegasubseteq mathbbC$ be the field of all constructible numbers (i.e. $Omega$ is the smallest subfield of $mathbbC$ which is closed under taking square roots). What is known about the automorphismgroup of the field $Omega$?
reference-request field-theory galois-theory profinite-groups
asked Apr 4 '15 at 11:36
user 59363
1,21539
add a comment |Â
up vote
9
down vote
favorite
up vote
9
down vote
favorite
Let $Omegasubseteq mathbbC$ be the field of all constructible numbers (i.e. $Omega$ is the smallest subfield of $mathbbC$ which is closed under taking square roots). What is known about the automorphismgroup of the field $Omega$?
reference-request field-theory galois-theory profinite-groups
asked Apr 4 '15 at 11:36
user 59363
1,21539
Let $Omegasubseteq mathbbC$ be the field of all constructible numbers (i.e. $Omega$ is the smallest subfield of $mathbbC$ which is closed under taking square roots). What is known about the automorphismgroup of the field $Omega$?
reference-request field-theory galois-theory profinite-groups
asked Apr 4 '15 at 11:36
user 59363
1,21539
asked Apr 4 '15 at 11:36
user 59363
1,21539
asked Apr 4 '15 at 11:36
user 59363
1,21539
asked Apr 4 '15 at 11:36
user 59363
1,21539
1,21539
add a comment |Â
add a comment |Â
1 Answer
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Every constructible field which is finite over Q has Galois group which is a 2-group. All finite 2-groups give constructible extensions of Q by a well-known theorem of Shafarevich. So the galois group you want is the inverse limit of all 2-groups.
answered Aug 25 at 0:43
i. m. soloveichik
3,58211125
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
Every constructible field which is finite over Q has Galois group which is a 2-group. All finite 2-groups give constructible extensions of Q by a well-known theorem of Shafarevich. So the galois group you want is the inverse limit of all 2-groups.
answered Aug 25 at 0:43
i. m. soloveichik
3,58211125
add a comment |Â
up vote
2
down vote
Every constructible field which is finite over Q has Galois group which is a 2-group. All finite 2-groups give constructible extensions of Q by a well-known theorem of Shafarevich. So the galois group you want is the inverse limit of all 2-groups.
answered Aug 25 at 0:43
i. m. soloveichik
3,58211125
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Every constructible field which is finite over Q has Galois group which is a 2-group. All finite 2-groups give constructible extensions of Q by a well-known theorem of Shafarevich. So the galois group you want is the inverse limit of all 2-groups.
answered Aug 25 at 0:43
i. m. soloveichik
3,58211125
Every constructible field which is finite over Q has Galois group which is a 2-group. All finite 2-groups give constructible extensions of Q by a well-known theorem of Shafarevich. So the galois group you want is the inverse limit of all 2-groups.
answered Aug 25 at 0:43
i. m. soloveichik
3,58211125
edited Aug 25 at 11:52
answered Aug 25 at 0:43
i. m. soloveichik
3,58211125
answered Aug 25 at 0:43
i. m. soloveichik
3,58211125
answered Aug 25 at 0:43
i. m. soloveichik
3,58211125
3,58211125
add a comment |Â
add a comment |Â
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