# Questions tagged [division-algebras]

A division algebra $D$ is a vector spaces over a field $F$ equipped with a bilinear product and a multiplicative neutral element $1$. All the non-zero elements of $D$ have a multiplicative inverse. Associativity is often assumed but not always. Any field is a commutative, associative division algebra. A skewfield = a division ring is always a division algebra over its center. The quaternions form the best known non-commutative division algebra.

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### direct products and direct sums of skew fields

If F is a skew field then are the arbitrary direct sums of F-modules isomorphic to their direct products (over the same index). I mean, if R is a division ring, and $\{M_i\}_{i\in I}$ some familly ...
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### “Vector spaces” over a skew-field are free?

Are modules over a skew field free? That is, if $F$ is a skewfield then can any module $M$ be written as $\underset{i \in I}{\bigoplus} F$ for some indexing set $I$?
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### Brauer group of a field of rational numbers

Can we say anything about Brauer group of $\mathbb{Q}$? And how can we construct it?
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### Is $SL_1(D)$ topologically finitely generated, for $D$ a division algebra over a local field?

I've been struggling with this one all day, and I was wondering if someone can give me a hand with the proof. I'm not even sure if the group in question is finitely generated, so I would appreciate if ...
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### normed division algebra

Can we prove that every division algebra over $R$ or $C$ is a normed division algebra? Or is there any example of division algebra in which it is not possible to define a norm? Definition of normed ...
In the article The Classification Of Real Division Algebras (written by R. S. Palais) it is said that if D is finite dimensional division algebra over $\mathbb{R}$, then: One way of stating the ...