Mixing
Proof of the main theorem on non-abelian Kummer extensions (following Lang)
I am trying to understand the proof of Theorem 11.1, Chapter VI from Lang’s Algebra and the conditions of Corollary 11.2. I have two specific questions:
- In the proof of 11.1, Lang says that the cocycle $cy_{sigma}$ should be trivial. It seems that he wants to use the fact that $[1+c]in G(A_N)$ and then apply Sah’s Lemma (Lemma 10.2). But his comments in the same section (I mean 11) imply that $[1+c]in G(A_M)$ if $M$ is not divisible by primes from $S$. Did I get it right? If yes, then why $[1+c]in G(A_N)$?
- In the conditions of Corollary 11.2, we require that $M$ is coprime to $2(Gamma’:Gamma)$. What is 2 factor for?
I appreciate any comments, may be someone knows a better/another reference on this topic.
galois-theory algebraic-number-theory group-cohomology
asked Nov 20 '18 at 20:40
Gregg
8818
add a comment |
I am trying to understand the proof of Theorem 11.1, Chapter VI from Lang’s Algebra and the conditions of Corollary 11.2. I have two specific questions:
- In the proof of 11.1, Lang says that the cocycle $cy_{sigma}$ should be trivial. It seems that he wants to use the fact that $[1+c]in G(A_N)$ and then apply Sah’s Lemma (Lemma 10.2). But his comments in the same section (I mean 11) imply that $[1+c]in G(A_M)$ if $M$ is not divisible by primes from $S$. Did I get it right? If yes, then why $[1+c]in G(A_N)$?
- In the conditions of Corollary 11.2, we require that $M$ is coprime to $2(Gamma’:Gamma)$. What is 2 factor for?
I appreciate any comments, may be someone knows a better/another reference on this topic.
galois-theory algebraic-number-theory group-cohomology
asked Nov 20 '18 at 20:40
Gregg
8818
add a comment |
I am trying to understand the proof of Theorem 11.1, Chapter VI from Lang’s Algebra and the conditions of Corollary 11.2. I have two specific questions:
- In the proof of 11.1, Lang says that the cocycle $cy_{sigma}$ should be trivial. It seems that he wants to use the fact that $[1+c]in G(A_N)$ and then apply Sah’s Lemma (Lemma 10.2). But his comments in the same section (I mean 11) imply that $[1+c]in G(A_M)$ if $M$ is not divisible by primes from $S$. Did I get it right? If yes, then why $[1+c]in G(A_N)$?
- In the conditions of Corollary 11.2, we require that $M$ is coprime to $2(Gamma’:Gamma)$. What is 2 factor for?
I appreciate any comments, may be someone knows a better/another reference on this topic.
galois-theory algebraic-number-theory group-cohomology
asked Nov 20 '18 at 20:40
Gregg
8818
I am trying to understand the proof of Theorem 11.1, Chapter VI from Lang’s Algebra and the conditions of Corollary 11.2. I have two specific questions:
- In the proof of 11.1, Lang says that the cocycle $cy_{sigma}$ should be trivial. It seems that he wants to use the fact that $[1+c]in G(A_N)$ and then apply Sah’s Lemma (Lemma 10.2). But his comments in the same section (I mean 11) imply that $[1+c]in G(A_M)$ if $M$ is not divisible by primes from $S$. Did I get it right? If yes, then why $[1+c]in G(A_N)$?
- In the conditions of Corollary 11.2, we require that $M$ is coprime to $2(Gamma’:Gamma)$. What is 2 factor for?
I appreciate any comments, may be someone knows a better/another reference on this topic.
galois-theory algebraic-number-theory group-cohomology
galois-theory algebraic-number-theory group-cohomology
asked Nov 20 '18 at 20:40
Gregg
8818
asked Nov 20 '18 at 20:40
Gregg
8818
asked Nov 20 '18 at 20:40
Gregg
8818
asked Nov 20 '18 at 20:40
Gregg
8818
asked Nov 20 '18 at 20:40
Gregg
8818
8818
add a comment |
add a comment |
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