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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What happens if in algebra we have several elements whose reciprocal is zero? [closed]

Suppose, we have in an algebraic system $\omega_1\ne\omega_2\ne\omega_3...$ and $1/\omega_1=0$, $1/\omega_2=0$, $1/\omega_3=0$,... What consequences should we expect? Is there a term for an element ...
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Divison algebra vs general/abstract algebra

I am not a mathematician but confused about division algebra vs. algebra. I suspect that "division algebra" is a sub-category(literally, not a math concept) of general or abstract algebra ...
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Since each $M_n(D)$-submodule of $M_n(D)$ is also a $D$-submodule, we have $M_n(D)$ artinian as a ring.

$D$ is a division ring so it is artinian, hence so is $M_n(D)$ as a $D$-module. Since each $M_n(D)$-submodule of $M_n(D)$ is also a $D$-submodule, we have $M_n(D)$ artinian as a ring. I am not sure ...
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Quotients of Two-sided and One-sided Ideals

Let $\mathcal{A}$ be a central simple algebra over an algebraic number field $K$, and $\mathcal{O}$ be a maximal $\mathcal{O}_K$-order in $\mathcal{A}$. Let $I$ be a maximal integral left-ideal of $\...
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63 views

Division automorphism

Let S be a set and A be a set of automorphisms of S such $\forall\ x,y\in S, \exists!\ a\in A \ |\ ax=y$ What's the name of this structure/property, that's similar to a "division algebra" ...
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Division Algebra with every element a root of $K[x]$

Let $D$ be a division algebra and $K\subseteq C(D)$ (the center of $D$). If every element of $D$ is a root of a nonzero polynomial in $K[x]$ prove that D is a field. I believe $D$ will end up being ...
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Find an algebraic division algebra that is not finite dimensional

I want to find an algebraic division algebra that is not finite dimensional, but i don't want to do it in terms of field extensions nor anything like that. Instead of that, what i want to do is to ...
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87 views

General dimension hairy ball theorem and division algebras

Question: Can someone please give a clear explanation, or point to a clear visual, that explains how the existence (or non-existence) of a non-vanishing continuous $n$-vector field on an $n$-sphere ...
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Proving that the Octonion Norm Respects Multiplication

I'm working with the following definition of the octonions: $\mathbb{O} = \mathbb{H} \times \mathbb{H}$, endowed with the product $$(p,q)(r,s) = (pr - sq^*, p^*s + rq).$$ Conjugation is given by $(p,q)...
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Uncommon notation for division algebra

I have found the following notion for a division algebra in a paper. $K=\mathbb{R}(x_1, \dots, x_n)$ is the field of rational functions in $n$ variables over $\mathbb{R}$ and $F=K((t))$ be the field ...
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Is every non-zero two-sided ideal of a polynomial ring over a division ring intersecting with the center?

Let $D$ be a division ring with center $F$, and $J$ a non-zero two-sided ideal of $D[x]$. Is it true that $J \cap F[x] \neq 0$? This is a question spawned from another problem I'm working on. And I ...
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For every element of an associative division $\mathbb{R}$-algebra there exists a quadratic equation with that element as a solution

If $D$ is an $\mathbb{R}$-algebra that is also a division ring and $\dim_{\mathbb{R}}D=n<\infty$, then for every $d\in D$ there exists $\lambda\in\mathbb{R}$ such that $d^2+\lambda d\in\mathbb R$. ...
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Bimodule over division algebras

Let $E_1$, $E_2$ be finite-dimensional division algebras over $\mathbb{Q}$. Let $X$ be a left $E_1 \otimes_\mathbb{Q} E_2^{op}-$module. In other words, $X$ is an $E_1-E_2-$bimodule, and $\mathbb{Q}$ ...
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Finite dimensional skew fields over $\mathbb{Q}$

Is there a specific reference for finite dimensional, associative, unital $\mathbb{Q}-$algebras that are division rings? Or also more in general, the type of questions I am trying to look into are: ...
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Normed Division Algebras: Why those dimensions? [duplicate]

I recently read a fascinating article about string theory, which discussed higher-dimensional algebras and their applications to supersymmery. The author mentioned that there were only four algebras ...
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Decomposition of a rank-deficient matrix

Given a complex or real $n\times m$ matrix $M$ with rank $r$, one can write it as $$M=LR$$ where $L$ is a $n\times r$ matrix, and $R$ is a $r \times m$ matrix. Does this hold for arbitrary fields? (I'...
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Does there exist a division ring without unity?

In abstract algebra I have only ever seen division introduced via multiplicative inverses, namely starting from a ring with unity $R$ and then adding the condition that each element $x$ has an inverse ...
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If $\Bbb R^n$ has a commutative division algebra structure over $\Bbb R$ then the multiplication map on $\Bbb R^n$ is continuous

I was reading the proof of the following theorem in Hatcher's Algebraic Topology: Theorem 2B.5. $\Bbb R$ and $\Bbb C$ are the only finite-dimensional divison algebra structure over $\Bbb R$ and have ...
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63 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 ...
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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 $k$?
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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: $\...
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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 ...
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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 ...
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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 ...
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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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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 $\mathbb{Q}_p$. ...
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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$_n(F)$. ...
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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?
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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 ...
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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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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 $$ (a,b) \times (c,d) = (ac+bd,ad-bc).$$ ...
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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 ...
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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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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 $\...
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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 ...
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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 ...
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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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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 ...
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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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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 $\...
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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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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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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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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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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}((...
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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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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 ...
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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 ...
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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 \...