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Questions tagged [brauer-group]

For questions about Brauer groups; in mathematics, the Brauer group of a field $K$ is an abelian group whose elements are Morita equivalence classes of central simple algebras over $K$, with addition given by the tensor product of algebras.

In mathematics, the Brauer group of a field $K$ is an abelian group whose elements are Morita equivalence classes of central simple algebras over $K$, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer.

The Brauer group arose out of attempts to classify division algebras over a field. It can also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras, or equivalently using projective bundles.

Source: https://en.wikipedia.org/wiki/Brauer_group

63 questions
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Is it really necessary to go to all this trouble to split division algebras?

Let $D$ be a finite-dimensional central simple algebra over a field $K$. $D$ is said to split over a field extension $L/K$ if we have $D_L \cong D \otimes_K L \cong M_n(L)$ for some $n$. In a few places (e.g. the Stacks Project) people seem to go to…
Qiaochu Yuan
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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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smooth curve of genus $0$ and Brauer group

In this blog of Alex Youcis, I see a sentence in the proof of theorem 4 which says that "since $C$ has genus 0 that it defines a class $[C]\in\mathrm{Br}(\mathbb{Q})$ which is trivial if and only if $C\cong\mathbb{P}^1_\mathbb{Q}$". And the…
user685167
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Construction of the Brauer Group

I've proven that the $K$ tensor product of two central simple $K$ algebras is itself central simple, and I've proven Wedderburn's theorem, but I now need to construct the Brauer group. I've been told that two algebras $A\cong M_n(D)$ and $B\cong…
Lammey
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Intuition behind the sum of $\mathrm{inv}$ maps in Albert-Brauer-Hasse-Noether theorem

I have been reading into class field theory lately and I would say, that I'm understanding, how the proofs for most of the theorems work, but I'm struggling a lot with the intuition. One of my big problems is the Albert-Brauer-Hasse-Noether theorem…
Firebolt2222
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Quaternion algebras over a non-Archimedean local field $K$, up to isomorphism

I want to know the number of non-isomorphic quaternion algebras over a non-Archimedean local field $K$. What is the number of non-isomorphic central simple algebras of dimension $n^2$ over a non-Archimedean local field $K$? I know the Brauer group…
Tireless and hardworking
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Field with vanishing Brauer group which is not $C_1$

In Serre's Local Fields he gives several examples of fields with trivial Brauer group. However, all of these examples are $C_1$ or conjectured to be $C_1$. Is there an example of a field which is not $C_1$ but has a trivial Brauer group?
user365205
5
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Degree $n$ extension of local field splits degree $n$ division algebra

I am trying to write an article which is pretty self-contained on the number theory side, and would like to use the following result: Let $K$ be a local field, $n > 1$ a natural number, $D$ a division algebra of degree $n$ over $K$. For any degree…
Bib-lost
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Brauer group of global fields

Is the Brauer group $\text{Br}(K)$ of a global field $K$ an $\ell$-divisible group for some prime $\ell$? If so, what $\ell$? Is $\text{Br}(K)[n]$ finite, for $n$ integer? I know from class field theory that it fits into an exact sequence $$0\to…
Julie
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On group graded algebras and Brauer groups

I was reading the paper "Algebras graded by groups" by Knus. I want to test and further my understanding of the paper by asking several questions. Since the paper is not readily available I will detail everything necessary to understand my…
Mathematician 42
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Hasse invariants under extension of scalars

Let $K\subset L$ be finite extensions of $\Bbb{Q}$. Background. Let $D$ be a finite dimensional division algebra with center $K$. Its class in the Brauer group $Br(K)$ then maps injectively into the direct sum of the Brauer groups $Br(K_v)$ with…
Jyrki Lahtonen
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Cohomology of local fields in positive characteristic

It is well-known from local class field theory that the Brauer group $\text{Br}(k)$ of a local field $k$ gets killed as you pass to sufficiently large extensions of $k$. In particular, $\text{Br}(L)(p) = 0$ for any extension $L/k$ of degree…
aspear
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Brauer Group for a Global Field with $l$-roots of unity $l\neq \text{char}(F)$

Let $F$ be global field that contains the $l$-roots of unity with $l$ a prime number different with the characteristic of $F$ and $\text{Br}F$ the Brauer Group of $F$. How can i proof that all element $\alpha$ in the $l$-torsion subgruoup…
Elvis Torres Pérez
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Direct product of group with itself mod diagonal subgroup

Let $G$ be any abelian group, and let $\triangle_G = \{(g,g)\mid g\in G\}\subset G\times G.$ Is there any significance in studying the quotient group ${(G\times G)}\left/{\triangle_G}\right.?$ If so, where does the study such an object naturally…
Chickenmancer
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Showing that a Severi-Brauer Variety with a point is trivial

Let $X/k$ be a variety over a field such that $X_{\overline k} \cong \mathbb P^n$ over $\overline k$ for some $n$. Suppose moreover that $X$ has a rational $k$-point $P$. Then, I know that $X \cong \mathbb P^n$ over $k$. The argument I know goes as…
Asvin
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