1
vote
2answers
81 views

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 ...
1
vote
1answer
47 views

Explicit Kummer isomorphism

Let $K$ be a characteristic $0$ field containing $\mu_n$ (the $n$-th roots of unity). Then it known that the map $K^{\times} / (K^{\times})^n \to \mathrm{Hom}(G, \mu_n)$ which sends $x$ to $\sigma ...
0
votes
0answers
43 views

First cohomology of a Galois group with finite base field

Let $l/k$ be a (may be infinite) galois extension with galois group $G$ and $k$ a finite field with size $q$. Also $k$ and $l$ are given the discrete topology. $G$ is given the Krull topology. Then ...
6
votes
0answers
69 views

Generalization of Kummer isomorphism?

Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action of $G:=Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ via the mod $p$ ...
2
votes
1answer
132 views

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 ...
1
vote
1answer
110 views

Lifting additive characters

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 ...
2
votes
2answers
258 views

Are absolute Galois groups compact topological groups

Let $T$ be a finite set of primes and let $K$ be the maximal extension of $\mathbf{Q}$ unramified outside $T$. We have three Galois groups: $G_{\mathbf{Q}} = ...