# 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.

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### Representations with traces in a finite field

Let $F$ be a finite field. Let $\rho:G\to GL_2(\overline F)$ be a continuous representation from a profinite group $G$, where $\overline F$ is an algebraic closure of $F$. Suppose that for all $g\in G$...
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### How does Wedderburns's little Theorem imply that central simple Algebras are isomorphic to matrix Algebras over a divison Algebra?

For a thesis I need the statement that every central simple algebra $A$ is isomorphic to $M_n(D)$ for some divison algebra $D$ and integer $n$. In a reference I am reading it states that this is a ...
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### When is a two-dimensional representation of a finite group over a finite field defined over a subfield?

I'm reading the proof of the Deligne-Serre theorem attaching Galois representations to newforms of weight one, and there's a representation-theoretic argument that I don't understand at all. The setup ...
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### Simple question about the Brauer group of a variety

If $X$ is a variety over a field $k$ and $\overline{k}$ is an algebraic closure of $k$, then there is a homomorphism $\text{Br}(X) \rightarrow \text{Br}(X \times_k \overline{k})$. Since $\text{Br}$ is ...
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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-...
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### Crossed products and division algebras

I am currently reading some introductory material on Brauer groups ("Noncommutative Algebra", by Farb and Dennis) and the following two questions came to my mind: 1) Are all crossed products algebras,...
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### Version of Wedderburn's theorem on central simple algebras

Suppose that $A$ be a central simple algebra over a field $k$. Then by Wedderburn's theorem $A\cong M_n(D)$ for some division $k$-algbera $D$. But to define the 'Brauer equivalence' I need that $D$ is ... ### If $L$ splits $D=I\otimes N$ does $L$ split $I$ and $N$?
Let $D$ be a central simple division algebra over the field $F$. Let $D\sim I\otimes N$ where $I$ and $N$ are division algebras in Br$(F)$ and $\sim$ is equivalence in Br$(F)$. I am interested in this ...