# Tagged Questions

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### multiple of a crossed homomorphism from finite group to a divisible one is principal

Let $\pi$ be a finite group, $\left|\pi\right|=n$ , acting on an abelian, torsion-free, $n$ -divisible group $D$ (i.e., every element of $D$ is divisible by $n$ ). Consider a crossed ...
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### Galois cohomology example

Let G be the Galois group of Q-bar over Q. I have a good idea of what $H^1(G,GL_1)$ is when $G$ acts trivially on $GL_1$: it is the group of homomorphisms from $G$ to $GL_1$. In particular it ...
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### Open subgroups $V\subset U$ of profinite group such that the index $[U:V]$ is divisible by $p$.

Let $G$ be a profinite group of $p$-cohomological dimension, $cd_p(G)\neq 0,\infty$. Consider an open subgroup $U\leq G$. Does there exist an open subgroup $V\leq U$ such that its index, $[U:V]$ is ...
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### Galois theory, had it solved any major problems beside its original applications to classical problems?

Galois theory, had it solved any major problems beside its original (classical) applications to roots of a fifth (or higher) degree polynomial equation (solvable algebraic equations and constructible ...
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### Classification of quadrtic forms over Q_p

I need some one to recap the topic with me and correct me when i am wrong. There are basically just a few questions at the end,but its important to also show what i know and not just what i dont. ...
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### Selmer and Shafarevich-Tate Groups

I'm currently trying to under the Selmer and Shafarevich-Tate Groups from Silverman's Arithmetic of Elliptic Curves (2nd edition), pg. 331 onwards. I have a couple of questions I think is derived from ...
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### Studying Galois Cohomology from Category Theory.

I am a masters student with background in Category Theory - I also know some Topos Theory. I would like to ask if you think it would be possible to start studying Galois Cohomology without any ...
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### $\operatorname{Br}(\Bbb Q_{47})$

I'm looking for division algebras over $\Bbb Q_{47}$. I guess my best bet is to calculate the Brauer group $\operatorname{Br}(\Bbb Q_{47})$. What's the best way of performing this calculation? Should ...
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### References for Composition Law on Binary Quadratic Forms

What are some good references for the Gauss composition law on binary quadratic forms in terms of Galois Cohomology? It is my understanding that there is a cohomological approach, and I am studying ...
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### Applications of additive version of Hilbert's theorem 90

Additive version of Hilbert's theorem 90 says that whenever $k \subset F$ is cyclic Galois extension with Galois group generated by $g$, and $a$ is element of $L$ with trace 0, there exists an ...
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### Galois groups of maximal unramified extensions

I have been workin throught Cassels-Frohlich Algebraic number theory. Im looking at local fields and their cohomology, in particular the Brauer group. On page 133 of casses-frohlich they make the ...
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### Why do the two topologies on a Galois group coincide?

In the following one is referred to the book. At this page, the author defines the Krull topology on a Galois group (not necessarily finite); on the 22nd page of the same book, the author defines the ...
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### Question on $\operatorname{res}^G_U \circ \operatorname{cor}^U_G = N_{G/U}$

This is probably a very basic question, but I can't wrap my head around it. Given a normal open subgroup $U$ of a profinite group $G$ and a $G$-Module $A$ we have the following equation in group ...
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### Local invariants of the discrete Galois module associated to a $p$-ordinary newform

Let $f=\sum_{n=1}^\infty a_nq^n$ be a $p$-ordinary newform of weight $k\geq 2$, level $N$, and character $\chi$, and let $\rho_f:G_\mathbf{Q}\rightarrow\mathrm{GL}_2(K_f)$ be the associated $p$-adic ...
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### On Grunwald-Wang theorem

Consider (roughtly speaking) the following statements (the Grunwald-Wang theorem) Theorem 1 (see here for details Wiki): Let $K$ be a number field and $x \in K$. Then under some conditions : $x$ is ...
Let $K$ a finite extension of $\mathbb{Q}_p$ ($p$ prime different from 2) and let $G_K$ the absolute Galois group of $K$. Let $\bar{u} : G_K \longrightarrow \mathbb{F}_p$ a continuous additive ...