# 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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### 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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### How do I prove that every fractional ideal of an order in a division algebra is a full lattice?

Let O be an R-order for some Dedekind domain R, let F be the field of fractions of R and D be a division algebra over F. A fractional left ideal of O is an R-lattice I in D such that OI in I (I ...
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### Why division ring has a center of a ring (=subring) is commutative and therefore division ring reflect itself a field?

I thought a ring was commutative for another reason but I realized that something I had not yet discovered, had led me to look for the solution in the wrong place. I see that 'commutative' property of ...
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### Dimension of division rings extension

Let $A \subseteq B$ be two division $k$-algebras, where $k$ is a field of characteristic zero. I am not sure if I wish to further assume that $B$ is affine over $A$, namely, if $B$ is finitely ...
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### The ring of quotients of the first Weyl algebra

Since there are no comments to this question, I now restrict it to the following question: It is known that: A ring $R$ is a prime left Goldie ring if and only if $R$ has a left quotient ring which ...
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### Cannot get the correct result following the division algorithm

I have the following algorithm which is supposed to be a Division algorithm: where m is a's number of digits and n is the b's number of digits. I tried to follow the steps taking a = 42 , b = 6 and ...
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### concept of conjugacy class in a ring

Can we think of a similar concept of a conjugacy class in a ring which satisfies two three properties like conjugacy classes. I think of a set $M_x={xyx^{-1}-y}$ for $x\in R$ and $R$ is a division ...
Suppose that $D$ is a division ring with center $F$ and with index $p$ prove that $D$ is cyclic if and only if there exists $x$ $\notin$$F which x$$^p$ $\in$$F. 0answers 103 views ### A corollary to the Wedderburn-Artin theorem. Suppose we proved the Wedderburn-Artin theorem, i.e. we have the fact that if S is a semisimple algebra over a field F, then$$ A \cong M_{n_1} (D_1) \times ... \times M_{n_k} (D_k),$$where$D_1,.....
Can someone proof the following? If $D$ is a division algebra over a field $F$, then as a $Z(D)$ algebra, $D$ is generated by the additive commutators (elements of the form $xy-yx$) Thanks!