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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Commutative subalgebras of a matrix algebra over a division algebra

Let $K$ be a field of characteristic $0$, $D$ a central semi-simple division algebra of dimension $d^2$ over $K$, and $n$ a positive integer. Let $R$ be a maximal subfield of $M_n(D)$, then is $\dim_K ...
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Skew fields with nonzero characteristic

There are many ways to construct a skew field of nonzero characteristic, e.g. the universal field of fractions of a skew polynomial ring $E[x;\sigma]$, a suitable choice of a quaternion algebra over $...
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Relationship between trivial Brauer group and commutative division algebras

Let $k$ be an algebraically closed field and $F$ a finite field extension of the field of rational functions $k(t)$. I've heard two different statements of Tsen's theorem under these conditions: ...
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Maximal subfield of the division algebra is splitting field

Currently I'm trying to get through "An Introduction to Algebraic K-theory" by C.A.Weibel. In the third chapter there is the following example. If $D$ is $d$-dimensional division algebra ...
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Fields with Finitely Many Division Rings

Let $\mathbb{k}$ be a field. For the purposes of this question, a division ring is a finite-dimensional $\mathbb{k}$-algebra $A$ in which every non-zero object is invertible. Is there a commonly ...
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The quadratic equation $x^2=c$ in a division ring

Let $D$ be a division ring. We denote $D'$ by the derived subgroup of the multiplicative group $D\setminus\{0\}$, that is, the subgroup generated by all the commutators of $D\setminus\{0\}$. For $c\in ...
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Can I tell from the group how the endomorphism rings of its representations will look like?

Let $G$ be a finite group. It is known that the endomorphism ring of an irreducible real representation $\rho: G\to\mathrm{GL}(\Bbb R^d)$ is (isomorphic to) either $\Bbb R$, $\Bbb C$ or $\Bbb H$ (the ...
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Examples of matrix groups with centralizer isomorphic to $\Bbb C$ or $\Bbb H$

By Schur's Lemma, the centralizer $C(G)$ of an irreducible matrix groups $G\subseteq\mathrm{GL}(\Bbb R^d)$ is an $\Bbb R$-division algebra, and thus, isomorphic to either $\Bbb R$, $\Bbb C$ or $\Bbb H$...
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Module viewed as a division algebra over its center

I am looking at an Artinian local ring, call it $(R,m)$. So R/m is a division ring. Denote by $F:= Z(R/m)$ its center. $F$ is evidently a field. Viewing $R/m$ as a division algebra over $F$, when ...
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Eichler orders in the quaternion.

Suppose $D$ be a quaternion over the number field $K$. $\Bbb{Definition.}$ The Eichler ordre $R$ of level $N$ in $D$ is defined as the one which satisfies the following equality with the completion $...
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when does a quaternion algebra isomorphic to $M_2(F)$?

We also suppose that the characteristic of a field is not $2.$ Definition 1. An algebra $B$ over $F$ is a quaternion algebra if there exist $i,j\in B$ such that $1,i,j,ij$ is an $F$-basis for $B$ ...
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On well-definedness of Shimura curve.

Suppose that we have a quaternion algebra $D$ over a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We assume that $D$ splits everywhere at finite places of $K$ and at ...
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Showing there exists no Division algebra on $\mathbb R^3$ indirectly (Sort of...)

Before you read my question please consider that i HAVE to do this exercise as i did below. So i am showing there exists no division algebra on $\mathbb R^3$. To show that, i have to show that for $*:...
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Desargues $\implies$ associativity: Projective planes over non-associative structures?

I've been reading about constructing projective planes over division rings (skewfields). There's this very nice fact that if Pappus's theorem holds in a division ring, this ring is actually ...
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An exercise about Division Algebra

In this, page 48, Exercies in chapter 1, there is a following exercise. Exercise 1. Let $D$ be a division algebra which has finite dimension over the field $k.$ For each $a\in D$ show there is a monic ...
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Splitting fields of subalgebras of central division algebras

Suppose $D$ is a central division algebra over $\mathbb{Q}$ of degree $n$. Let $A \subset D$ be a subalgebra, say also central over $\mathbb{Q}$ (to restrict the degree of generality). Now if a field $...
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A question to Finite Multiplicative subgroups in a division ring of I. N. Herstein

In this, I can't find the results in German as proof steps of Lemma 3 (... by Satz 88 [2, p. 72]) and Theorem 7 in page 123 (... Using results about division subalgebras of division algebras [1, p. 42,...
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"Canonical" norm on a real finite-dimensional unital associative division algebra?

Let $\mathcal{C}$ denote the category of unital associative finite-dimensional division $\mathbb{R}$-algebras. (As a full subcategory of that of unital $\mathbb{R}$-algebras.) For $A\in \mathcal{C}$ ...
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Prove $\mathbb{Q}$-subalgebra of quaternions is a division algebra over $\mathbb{Q}$

Let D be the $\mathbb{Q}$-subalgebra of $\mathbb{H}$ having basis $1, i, j, k$. Prove that D is a division algebra over $\mathbb{Q}$. This is an exercise from Rotman's Algebra. I only see that is to ...
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Division rings over fields

Given a field $K$. Rings are assumed to be with identity and associative. Question 1: Is there a (easy) construction of a non-commutative division ring with center $K$? Question 2: Is there a ...
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Affine domain over an algebraically closed field

An affine domain $A$ over $k$ is a finite dimensional $k$-algebra which is also an integral domain as a ring. Here's my thought. Fix $a \in A$ and $\varphi:k[x] \rightarrow A $ be defined as $f \...
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Could the discovery of a counterexample to the Unit Conjecture change mathematicians’ understanding of spinors in general?

Recently it has been reported that Giles Gardham has found a counterexample to the Unit Conjecture for group rings, as given in https://arxiv.org/abs/2102.11818 (and also in a popular article https://...
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$\mathbb{R}$ is closed under division

The set of real numbers $\mathbb{R}$ is closed under division. Does that mean $0$ is also considered? more specifically, should it be $\mathbb{R}-\{ 0 \}$? because division by $0$ is not defined.
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Uniqueness of the Eichler order

Let ${\mathrm{M}}_2(\widehat{{\cal O}_K})$ be the $2 \times 2$ matrices over the finite adele of the full integer ring ${\cal O}_K$ of a totally real #-field $K$. For the quaternion algebra $D_K$ over ...
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In a unital division algebra, does the unit have norm 1?

Suppose that $\mathbb{R}^n$ is a unital division algebra. That is, $\mathbb{R}^n$ is furnished with a bilinear map $$ *: \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^n $$ such that $$ x*y = 0 \...
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Fitting's Lemma version for pseudocompact modules or linearly compact modules

Let $R$ be a pseudocompact ring or a linearly compact ring. Is there a version of Fitting's Lemma for pseudocompoact or linearly compact $R$-modules?
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How can we represent a quaternion multiplication by quadrance and spread?

I don't speak well English, so please edit this question to be more accurate. Quaternions are considered as the quotient of the 3D vectors division. $${\bf v}\,{\bf r}^{-1}=-\frac{{\bf v}\,{\bf r}}{r^...
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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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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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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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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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