# 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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### 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 ...
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### 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 ...
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### Equivalent definitions of the Brauer group

I am trying to figure out a good way to see the equivalence in definitions of a brauer group of a field. The two are usually offered: either the brauer group has as elements, central simple algebras ...
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### Brauer groups, Milnor k theory and group cohomology

Can anyone suggest some basic material for learning connections between Brauer groups, Milnor k theory and group cohomology. I am an undergraduate. So, I find most of the sources available very hard.
A common proof that an algebraically closed field has trivial Brauer group goes something like this: Take $D$ a finite central division algebra over $K$, where $K$ is an algebraically closed field....
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 ...