# 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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### Surds and finding the answer for A & B

I have been given the following mathematic equation to solve for $A$ and $B$. I have gone through my knowledge of simplifying surds. Currently, I have gone down to: Square root $45 = 3\sqrt{5}$ Square ...
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### How does linear algebra over the octonions and other division algebras work?

An interesting question, which has been discussed in many forms on this site, is how many results from the study of linear algebra over vector spaces carries over when we allow the scalars to form an ...
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### Computing the inverse of a full lattice in a quaternion algebra

Let $D$ be quaternion algebra over a number field $F$. Let $\Delta\subseteq D$ be a maximal $\mathcal{O}_{F}$-order. Let $\mathfrak{b}$ be a fractional left $\Delta$-ideal. In his book "Maximal Orders"...
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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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### Which number fields can appear as subfields of a finite-dimensional division algebra over Q with center Q?

I have some idle questions about what's known about finite-dimensional division algebras over $\mathbb{Q}$ (thought of as "noncommutative number fields"). To keep the discussion focused, let's ...
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### Central simple algebras give rise to algebraic groups

Let $D$ be a central division algebra over a field $k$ of dimension $n^2$. I have heard that the functor $$R \mapsto (D \otimes_k R)^{\ast}$$ going from commutative $k$-algebras to groups is ...
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### Subfields of central division algebras over fixed global field

A generalization of this question: Let $K$ be a global field, could any finite field extension of $K$ be embedded in a finite dimensional central division algebra over $K$? The answer is true locally....
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### Is it true that a division algebra as a module over itself is a simple module?

If we have a division algebra $A$, is it just a simple module over itself? Given a submodule $B$ of $A$ and $b \in B$, $\exists$ $b^{-1} \in B:bb^{-1}=1 \in B$, and so $B = A$. Is this argument ...
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### Classification of a certain family of real division algebras

Let $A$ be a (non-commutative, not nessasarilly associative) division algebra over $\mathbb{R}$ such that $\mathbb{R}^3 \subset A$. Assume that for any two nonzero vectors $u, v \in \mathbb{R}^3$ we ...
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### Conditions for equivalent definitions of division algebra

A division algebra is defined as a (not necessarily finite dimensional, associative, or unital) algebra $A$ over a field, where $\forall a\neq0,b\in A$ the equations $ax=b$ and $ya=b$ have unique ...
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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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### Proving that $\mathbb R^3$ cannot be made into a real division algebra (and that extending complex multiplication would not work)

I am trying to solve the following exercise: Prove that complex multiplication does not extend to a multiplication on $\mathbb R^3$ so as to make $\mathbb R^3$ into a real division algebra. I ...
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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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### Why does the multiplication in a division algebra depends on every component?

In a division algebra A over $\mathbb{R}$ we have this multiplication (A isomorphic to $\mathbb{R}^{n}$) $$\mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}^{n}:(x,y)\mapsto y=x\cdot y$$ where every ...
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### Prove that the homomorphism $\phi:R\to S^{-1}R$ is injective if and only if $S$ contains no zero-divisors. [duplicate]

Suppose that $S\subset R$ is a multiplicative set in $R$, where $R$ is a commutative ring with identity $1\neq 0$. Prove that the homomorphism $\phi:R\to S^{-1}R$ is injective if and only if $S$ ...
Let $D$ be a division ring and let $K$ be the center of $D$. Assume $\dim_K(D)<\infty$. Why is $\dim_K(D)$ a square?