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Questions tagged [galois-cohomology]

For questions on Galois cohomology, the study of the group cohomology of Galois modules.

In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor.

241 questions
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Projective profinite groups

I'm reading the first chapter of Serre's Galois Cohomology. On p. 58, He gives two examples of projective profinite groups: the profinite completion of free (discrete) groups; the cartesian product over all prime numbers of free pro-p…
Immanuel
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Relationship between Galois cohomology and etale cohomology.

Why is étale cohomology a natural generalization of Galois cohomology ? I would like to inform you that I have a few quite sufficient prerequisites Galois cohomology and its application to solve the $90$ - th problem of Hilbert . So I can…
Albert007
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ch. 8.3 Exercise 1 in Cohomology of Number Fields

Let $k$ be a number field, $S$ a set of places of $k$, $k_S$ the maximal extension of $k$ unramified outside $S$, $\mathcal{O}_S$ the subring of $k_S$ with $\nu_{\mathfrak{p}}(\alpha)\geq 0$ for all $\mathfrak{p}\not\in S$, and $G_{S}$ the galois…
Snacc
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Which number fields allow higher genus curves with everywhere good reduction

The field of rational numbers is not such a number field. That is, there does not exist a smooth projective morphism $X\to\text{Spec } \mathbf{Z}$ such that the generic fibre is a curve of genus $\geq 1$. Which number fields allow (or do not allow)…
Soka
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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 $p$ cyclotomic character. We have a nice description of…
user33624
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Computing étale cohomology group $H^1( \text{Spec}(k), \mu_n)$ and $H^1( \text{Spec}(k), \underline{\Bbb{Z}/\mathord{n \Bbb{Z}}})$

I am starting to learn about étale cohomology and would like to compute a simple example. Let $k$ be a field with a fixed separable extension $k^s.$ I want to compute $H^1( \operatorname{Spec}(k), \mu_n)$ (I am speaking about the étale cohomology…
proofromthebook
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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 an $n$-th power iff $x$ is an $n$-th power almost…
user10676
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$\sqrt{2} \notin \mathbb{Q}$ and Galois Cohomology

If I look up infinite descent on Wikipedia we get the sample of proving that $\sqrt{2} \notin \mathbb{Q}$ -- it is irrational. $$\sqrt{2} = \frac{a}{b} \to 2 = \frac{a^2}{b^2} \to 2b^2 = a^2 \to 2b^2 = (2c)^2 = 4c^2 \to b^2 = 2c^2 $$ I apologize for…
cactus314
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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 element $b$ of $L$ such that $a = b - g(b)$. The…
xyzzyz
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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 polygons) ? I understand that Galois Theory had…
ben
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some questions about the Robba ring

Notations and definitions Let $p$ be a prime integer, $k$ be a perfect field of characteristic $p$ and $W(k)$ its ring of Witt vectors. Definition 1 We put $$ \mathcal{R}_r=\bigg\{ \sum_{i\in \mathbf{Z}}a_iu^i: a_i\in W(k)[1/p], \lim_{i\to \pm…
lzhao
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Motivation for the relations defining $H^1(G,A)$ for non-commutative cohomology

First let me review the definition of first non-commutative cohomology. Let $G$ be a group and $A$ a left $G$-group, i.e. for any $\sigma, \tau\in G$ and $a, b\in A$, one has $\sigma(\tau(a))=(\sigma\tau)(a), \sigma(ab)=\sigma(a)\sigma(b)$. A…
minimax
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Why is this cubic polynomial generic for cyclic field extensions?

[EDIT: There doesn't seem to be any interest in answering this question, so could anyone just provide me a reference for understanding (2), and if possible (1)? Hopefully that would be enough to help me move forward on this.] On page 1 of Serre's…
Somatic Custard
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Why are central simple algebras classified by cohomology?

In their article on the Brauer group Wikipedia writes: Since all central simple algebras over a field $K$ become isomorphic to the matrix algebra over a separable closure of $K$, the set of isomorphism classes of central simple algebras of degree…
John C. Baez
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Question about inertia groups and unramified extensions

Let $K$ be a number field, and $v$ a finite place. If $\bar{K}$ is a separable closure of $K$, then in $G_K=\text{Gal}(\bar{K}/K)$ we can find the decomposition group of (a place over) $v$, which is isomorphic to the Galois group of $\bar{K_v}/K_v$,…
Lios
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