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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105
votes
6answers
6k views

Why are the solutions of polynomial equations so unconstrained over the quaternions?

An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb C$,...
1
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1answer
41 views

Are all central simple algebras unital?

The definition I'm using for a CSA over a field $k$ is the following: A CSA over $k$ is a finite-dimensional associative $k$-algebra which is simple and has center precisely $k$. My question ...
25
votes
5answers
7k views

What are some real-world uses of Octonions?

... octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative. Comes from a a quote by John Baez. Clearly, the sucessor to quaterions from the Cayley-Dickson process is ...
3
votes
1answer
60 views

Maximal central subalgebra(s) in non-central division $k$-algebra

Let $D$ be a finite dimensional (non-central) division $k$-algebra, where $k$ is a field. Is there a concrete description of the maximal subalgebra(s) of D having center $k$?
5
votes
1answer
134 views

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 ...
0
votes
0answers
21 views

Commutation or Anti-commutation of the corresponding imaginary units of the octonions and split-octonions

Given a general octonion x: $\mathbb{O}$=$\mathbb{H}$+$\mathbb{H}$$L$ by x=$x^1$+$x^2$i+$x^3$j+$x^4$k+$x^5$i$L$+$x^6$j$L$+$x^7$k$L$+$x^8$$L$ with $L^2$=-1, and a general split-octonion x: $\...
2
votes
1answer
116 views

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 ...
2
votes
1answer
37 views

Identity in a composition algebra

Let $A$ be a real composition algebra ($A=\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$). I would like to prove that $$ |\lambda|=1 \implies(\lambda u) \overline{(\lambda v)}=u\overline{v}$$ In a ...
1
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1answer
267 views

Indefinite quaternion algebra over Q

Let $D$ be an indefinite quaternion algebra over $\Bbb Q$. We have a chosen isomorphism $\iota \colon D \otimes_{\Bbb Q} {\Bbb R} \cong {\mathrm{M}}_2({\Bbb R})$. Q: If we choose another isomorphism $...
7
votes
2answers
115 views

What does $(a,b)_{\zeta}$ correspond to in $\mathrm{Br}(\mathbb{Q}_p)=\mathbb{Q}/\mathbb{Z}$

Let $p$ be a prime number, let $\mathbb{Q}_p$ be the local field, by Hensel's lemma, we know it has $p-1$-th roots of unity, let $\zeta$ be a fixed primitive $p-1$-th root of unity in $\mathbb{Q}_p$. ...
1
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0answers
70 views

Maximal subfield of a central simple algebra which is not Galois

In the book Algebra IX: Finite Groups of Lie type and Finite Dimensional Algebra, the authors Kostrikin-Shafarevich mention (p. 159) that If $A$ is a central simple algebra over $F$ of finite ...
2
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0answers
26 views

Invertible elements of CSA inducing Galois automorphism are linearly independent

Let $A$ be a central simple algebra over a field $F$. Let $K$ be a maximal subfield of $A$ with $[K:F]=n$ and assume $K$ is Galois extension of $F$. Let $\sigma_1,\sigma_2,\cdots,\sigma_n$ be all ...
4
votes
0answers
80 views

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; $D$ ...
3
votes
0answers
98 views

(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 ...
6
votes
1answer
139 views

Dimension of $End(V)$ with $V$ countable dimension irreducible module over a complex algebra

Let $A$ be a $\mathbb{C}$-algebra and $V$ be an irreducible $A$-module with countable dimension. What is the dimension of $End(V)$ as $A$-module? Note that every endomorphism must be injective and ...
0
votes
0answers
98 views

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, ...
1
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0answers
39 views

Subdomains of matrix algebras

Let $F$ be a field and M$_n(F)$ the ring of $n\times n$ matrices. By a domain we mean a not necessarily commutative ring without zero divisors. We consider subdomains $R$ of the ring M$_n(F)$. ...
2
votes
1answer
32 views

Dimension of $\operatorname{End}_{\mathbb C} \mathbb H$ as $\mathbb {R}$ vector space.

On page 15 of this note $\operatorname{End}_{\mathbb C} \mathbb H$ is an $8$-dimensional real vector space. Is there a simple way to see this?
2
votes
1answer
222 views

Finite dimensional division algebras over the reals other than $\mathbb{R},\mathbb{C},\mathbb{H},$ or $\mathbb{O}$

Have all the finite-dimensional division algebras over the reals been discovered/classified? The are many layman accessible sources on the web describing different properties of such algebras, but ...
2
votes
1answer
89 views

Unital nonalternative real division algebras of dimension 8

The finite-dimension division algebras over the reals are: $\Bbb R$: the reals (dimension 1) $\Bbb C$: the complex numbers (dimension 2) $\Bbb H$: the quaternions (dimension 4) $\Bbb O$: the ...
3
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0answers
108 views

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 ...
21
votes
1answer
535 views

What is this 2D division algebra?

Consider the set $A$ of 2-tuples of real values $(a,b)$, equipped with an addition defined as $$ (a,b) + (c,d) = (a+c,b+d)$$ and multiplication defined as $$ (a,b) \times (c,d) = (ac+bd,ad-bc).$$ ...
3
votes
1answer
134 views

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 ...
3
votes
1answer
55 views

Algebraic division algebra over Euclidean field

By Frobenius' Theorem we know that if $D$ be an algebraic non-commutative division algebra over $\mathbb{R}$ then ,as an $\mathbb{R}$-algebra, $D$ is isomorphic to $\mathbb{H}$. We can also replace $\...
1
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0answers
39 views

Central simple algebra of dimension 4

Suppose $A$ is a $F$-central simple algebra with maximal subfield $E$ such that $[A:F] = 4$. if $N_{E/F}(E^*) \ne F^*$, then $A$ is a division algebra. Is this even true? If it is true how i can ...
4
votes
1answer
646 views

Is every endomorphism of the quaternion ring surjective?

Is the quaternion ring an EAS Division ring? An EAS Division ring is a ring $D$ such that each endomorphism of $D$ is surjective. I know that $\mathbb{R}$ and $\mathbb{Q}$ are EAS Division rings.
2
votes
1answer
64 views

Splitting fields of a divison algebra

Let $k$ be a field, $D$ be a central division algebra of degree $n$ over $k$. We call $k'$ a splitting field of $D$ if $D\otimes_kk'\cong M_n(k')$. Splitting fields may not be isomorphic, can we say ...
0
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0answers
42 views

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 ...
1
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0answers
60 views

Are all quaternion algebras over the rationals skew fields?

If I understand correctly, any quaternion algebra over the rationals is a noncommutative associative division algebra. I am currently working with implementations of quaternion algebras in MAGMA and ...
3
votes
2answers
536 views

Simple $M_n(D)$-module with $D$ a division ring

Define $D$ to be a division algebra over a field $k$ and $R=M_n(D)$ the $n\times n$ matrix ring over $D$. A simple $R$-module $M$ is the quotient of $R$. I can write $R=\bigoplus_j I_j$ where $I_j$ is ...
0
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0answers
36 views

Solving linear systems for integer values in MAGMA

Say we are given a quaternion algebra D over a number field F as well as a maximal $\mathcal{O}_F$-order $\Delta$ $\subseteq$ D and say we have a $\mathbb{Z}$-basis $\omega_1, . . . , \omega_n$ for $\...
1
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0answers
55 views

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"...
0
votes
0answers
84 views

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 ...
32
votes
1answer
743 views

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 ...
4
votes
1answer
111 views

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 ...
2
votes
1answer
45 views

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....
1
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1answer
29 views

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 ...
1
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0answers
75 views

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 ...
3
votes
0answers
98 views

Brauer group of the field of Laurent series with coefficients in a finite field

In a course I attended at university, we calculated the Brauer group of $\mathbb{F}_q((t))$ with $q=p^n$ , $p$ prime number and we proved it was $\dfrac {\mathbb{Q}}{\mathbb{Z}}=Br(\bar{\mathbb{F}_q}((...
0
votes
1answer
33 views

Central simple quaternion algebra: why is the matrix for $\rho(v)$ antidiagonal?

Let $F$ be a field of characteristic $0$. Let $D$ be a central, simple quaternion division algebra over $F$. Let $x \in D$, not in $F$. Then $K = F[x]$ is a field of degree two over $F$, and $D$ is ...
2
votes
0answers
108 views

The ring generated over the center of a division ring by a group.

Let $D$ be a division ring with the center $F$. Suppose that $G$ is a subgroup of the multiplicative group of $D$ such that every element of $G$ is algebraic over $F$. Then may we conclude that any ...
6
votes
2answers
201 views

Finite dimensional central division $\mathbb K$-algebra as a subalgebra of a matrix $\mathbb K$-algebra

The question is as follows: A finite-dimensional central division $\mathbb K$-algebra $D$ is a $\mathbb K$-algebra isomorphic to a subalgebra of $M_r(\mathbb K)$ if and only if $\dim_{\...
0
votes
0answers
35 views

How to construction a division ring from the given field?

Let $F$ be a field of characteristic 2. How could we construct a division ring $D$ which centre is $F$. Where division ring mean non-commutative ring with unity $1$ and for each non-zero element $x \...
2
votes
2answers
70 views

Existence of division algebras with center $\mathbb{Q}$ of prime degree

In First Course in Noncommutative rings of T.Y.Lam (p.210), the author stated that "It is known that for each $n$, there exists a $\mathbb{Q}$-division algebra $A_n$ of dimension $p_n^2$, with $Z(A_n)=...
2
votes
0answers
45 views

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 ...
5
votes
0answers
54 views

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 ...
11
votes
2answers
1k views

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 ...
1
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0answers
27 views

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 ...
0
votes
0answers
150 views

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 $S$ ...
3
votes
2answers
166 views

The dimension of a division ring over its center is square. [closed]

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?