# 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.

52 questions
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### Do Hopf bundles give all relations between these “composition factors”?

Write a fiber bundle $F\to E\to B$ in short as $E=B\ltimes F$ (in analogy with groups). (This is not necessary, but: given another bundle $X\to B\to Y$, we can write $E=(Y\ltimes X)\ltimes F$, but ...
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### Galois theory for (non-commutative) division rings

Is there a 'Galois theory' with fields replaced by (non-commutative) division rings? I have googled this, and it seems that there are known results in that direction, for example, this paper which ...
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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 ...
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### finite division algebras over a field

A theorem of Wedderburn says that finite division ring is field. Here, ring means "ring with unity and is also associative". In particular, finite division (associative) algebras are fields. I was ...
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### 4-D lattices and quaternions

It is easy to prove that there are only 2 extensions $\mathbb{Q}(a)$, with $|a|=1$, of $\mathbb{Q}$ where $\mathbb{Z}[a]$ becomes a lattice(discrete free abelian subgroup of rank 2) in the complex ...
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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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### Division algebra over rationals of dimension 9

I want to understand about existence of some non-commutative division algebras over $\mathbb{Q}$ of dimension $9$. Q. Does there exist a division algebra $D$ such that $D$ is non-commutative; $D$ ...
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### (Non-Associative Division Algebras) Can someone help me find where the contradiction is?

This has been bugging me for a while any help would be appreciated. The second bullet point from here says: Let A be a non-associative unital algebra with finite dimension, then it's possible to ...
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### algebraically closed for more than simple polynomials

Is there any non-trivial algebra for which any non-constant algebraic expression has a root in that algebra? For example the complex numbers have a solution for any basic polynomial, but do not have ...
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### Weighted Division with more than one factors

Let's say I have 6,000 dollars and I want to divide them in 5 different companies. The logic thing is to devide the 2 amounts and find the money per company. Now let's say I have 3 characteristics for ...
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### tensor product of division algebras

Let $\Delta_{1}$ and $\Delta_{2}$ be finite dimensional division algebras over field $F$ and $\Delta_{1}$ is central, then $\Delta_{1}\otimes\Delta_{2} = M_{r}(E)$ where $E$ is a division algebra, ...