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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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Weighted Division with more than one factors

Let's say I have 6,000 dollars and I want to divide them in 5 different companies. The logic thing is to devide the 2 amounts and find the money per company. Now let's say I have 3 characteristics for ...
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1answer
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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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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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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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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, ...
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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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Corestriction of algebras and isomorphism

Let $L/K$ be finite and separable and $F/K$ an arbitrary extension and $E=L\otimes F$ be a field. If $A$ be $L$-algebra. is it true that we have $F$-algebra isomorphism: $c_{L/K}(A)\otimes F\simeq c_{...
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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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1answer
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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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1answer
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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 \...
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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_{\...
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2answers
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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)=...
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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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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 $S$ ...
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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?
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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 ...
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If $R$ is a finitely generated $D$-algebra, $D$ a division ring, then $R$ has finite basis over $D$?

Let $k \subset D \subset R$, where $k$ is a field of characteristic zero, $D$ is a division $k$-algebra, and $R$ is affine over $D$ (= $R$ is a finitely generated $D$-algebra). $R$ is a free $D$-...
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Let $D$ be a division ring. Show that if every $a \in D$ is algebraic over the prime subfield of $D$ then $D$ is commutative [closed]

Let $D$ be a division ring. Show that if every $a \in D$ is algebraic over the prime subfield of $D$ then $D$ is commutative ($D=Z(D)$).
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Let $D$ be a countably dimensional division algebra over an uncountable algebraically closed field $F$. Why $D=F$?

Problem. Suppose that $F$ is an uncountable algebraically closed field and $D$ is a division $F$-algebra. If $\dim_F D$ is countable, then $D=F$. For each $x\in D$, $F(x)$ is a field extension of $F$ ...
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1answer
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Fields for $\mathbb Q^3$?

I am just a happy beginner in anything algebraic. It is discussed in this question why no fields exist for $\mathbb R^3$, but what about $\mathbb Q^3$? Can we build a division algebra by excluding the ...
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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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The division ring of fractions of the first Weyl algebra and its subrings

The first Weyl algebra, $A_1(k)= k\langle x,y | yx-xy=1\rangle$, where $k$ is a field of characteristic zero, is known to be a simple Noetherian ring, hence it has a (left) division ring of fractions (...
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1answer
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Multiplicativity formula for a tower of division algebras

The multiplicativity formula for degrees of a tower of fields is well-known. I wonder if the same formula still holds if we consider division rings instead of fields, namely: Let $A \subseteq B \...
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1answer
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Primitive element theorem for division rings extension

Assume that $A \subseteq B$ are division $k$-algebras, where $k$ is a field of characteristic zero. Further assume that $A$ has the IBN property (does a division ring 'automatically' have the ...