All Questions
Tagged with class-field-theory gr.group-theory
9 questions
4
votes
1
answer
246
views
How do "Kummer closures" of fields look?
Let $F$ be a field and $A$ a finite abelian group. You can ask: does the regular representation $F[A]$ of $A$ split as a direct sum of 1-dimensional representations? This is equivalent to the ...
- 54.6k
0
votes
1
answer
454
views
Coboundary operators, 1-cocycles and computing cohomology
My question is about the compatibility and consistency between two definitions of cohomology in two books.
I asked this question about 10 days ago on MathSE
and I set a bounty on it, but I didn't ...
0
votes
0
answers
124
views
When is the natural map of Tate cohomology an isomorphism?
First of all I want to say that I am not at all an expert in Group cohomology . Recently I attended a seminar where the speaker mentioned about something called Tate cohomology groups which in ...
- 1,215
7
votes
2
answers
1k
views
Galois groups and prescribed ramification
What is known about finite groups $G$ for which there exists a Galois extension $K$ of $\mathbb{Q}$ ramified only at $2$ such that $\text{Gal}(K/\mathbb{Q}) \cong G$ ? More generally, which groups can ...
- 11.3k
12
votes
1
answer
1k
views
Class field towers
It is known (Golod and Shafarevich) that the class field tower of a finite extension $K$ of $\mathbb{Q}$ may be infinite. But is it always finite for $K=\mathbb{Q}[\zeta]$ where $\zeta$ is a root of ...
user6976
6
votes
0
answers
293
views
Iwasawa theory, $\mathbb{Z}_p^{2}$-extension, Greenberg module
Take $H\subset \bar{\mathbb{Q}}$ be a quartic imaginary number field such that $\operatorname{Gal}(H/\mathbb{Q})=\mathbb{Z}_2 \times \mathbb{Z}_2$. Denote by $F$ the quadratic real subfield of $H$ and ...
- 1,066
9
votes
2
answers
882
views
transfer kernels and the Schur multiplier
Let $\Gamma$ be a finite $2$-group, and let $G$ be any subgroup
of index $2$. Moreover, let Ver$: \Gamma/\Gamma' \to G/G'$
denote the group theoretical transfer, and let $M(\Gamma)$ be
the Schur ...
- 32.6k
6
votes
0
answers
221
views
Furtwangler's Principal ideal theorem in number fields
Does anyone know a simple proof, using cohomological method of the fact that the verlagerung from a finite group G. to its commutator subgroup G', i.e. $$G/G'->(G')^{ab}$$ vanishes?
The simplest ...
- 61
4
votes
1
answer
491
views
group theoretical transfer map and its consequences
I'm trying to understand whether there is a sophisticated reason that forces the transfer map to play its role in class field theory or not. Because, at least in Neukirch's proof (at his book ANT) on ...
- 295