Prove or disprove: Two different subgroups of a Galois group will have different fixed fields.
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The Fundamental Theorem of Galois Theory clearly doesn't hold since it doesn't specify that the groups are finite, so I'm not sure how to proceed.
galois-theory
asked Sep 6 at 3:18
user6983
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The Fundamental Theorem of Galois Theory clearly doesn't hold since it doesn't specify that the groups are finite, so I'm not sure how to proceed.
galois-theory
asked Sep 6 at 3:18
user6983
1
If you have wandered into the realm of infinite Galois extensions then the Fundamental Theorem of Galois Theory states that the intermediate fields are the fixed fields of the closed subgroups, so it looks like you need to exhibit a subgroup that is not closed (the closure will share the fixed field).
– Jyrki Lahtonen
Sep 6 at 5:18
As you are new I want to make sure that you looked at the guide for new askers.
– Jyrki Lahtonen
Sep 6 at 5:20
And, perhaps, you should take a peek at this.
– Jyrki Lahtonen
Sep 6 at 5:33
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
The Fundamental Theorem of Galois Theory clearly doesn't hold since it doesn't specify that the groups are finite, so I'm not sure how to proceed.
galois-theory
asked Sep 6 at 3:18
user6983
1
The Fundamental Theorem of Galois Theory clearly doesn't hold since it doesn't specify that the groups are finite, so I'm not sure how to proceed.
galois-theory
galois-theory
asked Sep 6 at 3:18
user6983
1
asked Sep 6 at 3:18
user6983
1
asked Sep 6 at 3:18
user6983
1
asked Sep 6 at 3:18
user6983
1
asked Sep 6 at 3:18
user6983
1
1
If you have wandered into the realm of infinite Galois extensions then the Fundamental Theorem of Galois Theory states that the intermediate fields are the fixed fields of the closed subgroups, so it looks like you need to exhibit a subgroup that is not closed (the closure will share the fixed field).
– Jyrki Lahtonen
Sep 6 at 5:18
As you are new I want to make sure that you looked at the guide for new askers.
– Jyrki Lahtonen
Sep 6 at 5:20
And, perhaps, you should take a peek at this.
– Jyrki Lahtonen
Sep 6 at 5:33
add a comment |Â
If you have wandered into the realm of infinite Galois extensions then the Fundamental Theorem of Galois Theory states that the intermediate fields are the fixed fields of the closed subgroups, so it looks like you need to exhibit a subgroup that is not closed (the closure will share the fixed field).
– Jyrki Lahtonen
Sep 6 at 5:18
As you are new I want to make sure that you looked at the guide for new askers.
– Jyrki Lahtonen
Sep 6 at 5:20
And, perhaps, you should take a peek at this.
– Jyrki Lahtonen
Sep 6 at 5:33
If you have wandered into the realm of infinite Galois extensions then the Fundamental Theorem of Galois Theory states that the intermediate fields are the fixed fields of the closed subgroups, so it looks like you need to exhibit a subgroup that is not closed (the closure will share the fixed field).
– Jyrki Lahtonen
Sep 6 at 5:18
If you have wandered into the realm of infinite Galois extensions then the Fundamental Theorem of Galois Theory states that the intermediate fields are the fixed fields of the closed subgroups, so it looks like you need to exhibit a subgroup that is not closed (the closure will share the fixed field).
– Jyrki Lahtonen
Sep 6 at 5:18
As you are new I want to make sure that you looked at the guide for new askers.
– Jyrki Lahtonen
Sep 6 at 5:20
As you are new I want to make sure that you looked at the guide for new askers.
– Jyrki Lahtonen
Sep 6 at 5:20
And, perhaps, you should take a peek at this.
– Jyrki Lahtonen
Sep 6 at 5:33
And, perhaps, you should take a peek at this.
– Jyrki Lahtonen
Sep 6 at 5:33
add a comment |Â
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If you have wandered into the realm of infinite Galois extensions then the Fundamental Theorem of Galois Theory states that the intermediate fields are the fixed fields of the closed subgroups, so it looks like you need to exhibit a subgroup that is not closed (the closure will share the fixed field).
– Jyrki Lahtonen
Sep 6 at 5:18
As you are new I want to make sure that you looked at the guide for new askers.
– Jyrki Lahtonen
Sep 6 at 5:20
And, perhaps, you should take a peek at this.
– Jyrki Lahtonen
Sep 6 at 5:33