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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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Do Hopf bundles give all relations between these “composition factors”?

Write a fiber bundle $F\to E\to B$ in short as $E=B\ltimes F$ (in analogy with groups). (This is not necessary, but: given another bundle $X\to B\to Y$, we can write $E=(Y\ltimes X)\ltimes F$, but ...
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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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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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4-D lattices and quaternions

It is easy to prove that there are only 2 extensions $\mathbb{Q}(a)$, with $|a|=1$, of $\mathbb{Q}$ where $\mathbb{Z}[a]$ becomes a lattice(discrete free abelian subgroup of rank 2) in the complex ...
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Crossed products and division algebras

I am currently reading some introductory material on Brauer groups ("Noncommutative Algebra", by Farb and Dennis) and the following two questions came to my mind: 1) Are all crossed products algebras,...
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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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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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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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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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Are division algebras over local fields compact mod center?

If D is a central division algebra over a local field F, is it true that $D^\times/F^\times$ is compact?
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A Condition for division ring being a field.

I'm reading a paper of Maurice Chacron and I.N. Herstein entitled Powers of skew and symmetric elements in division rings. At the first page of the paper, I got stuck in a problem that: "if in a ...
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Existence of a division ring on a field.

Suppose that $F$ is a field. Show that there exists an $F$-division algebra $D$ with two elements $a\neq b\in D$ such that $a^2-2ab+b^2=0$. In the field extensions we know that $a^2-2ab+b^2=0$ if and ...
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Is the generalized hopf map of an alternative finite-dimensional real division algebra continuous?

Let $A$ be an $n$-dimensional alternative real division algebra (not necessarily associative). Is the map $$ \eta \colon \bigl\{(x,y) \in A \times A : |x|^2+|y|^2=1\bigr\} \to A \sqcup \{\infty\}, \...
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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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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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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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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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Indefinite unitary group over split quaternions

Denote by $\mathbb{C}_{\mathrm{sp}}$ be the split complex numbers. This is isomorphic to the direct sum $\mathbb{R}\oplus\mathbb{R}$ with norm $N(a,b)=ab$ and conjugation $\overline{(a,b)}=(b,a)$. ...
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Motivating the Cayley-Dickson construction by proving Hurwitz's theorem

To me it seems the way to motivate the Cayley-Dickson construction is to prove Hurwitz's theorem, which is done over at Wikipedia. The theorem states the only real division algebras equipped with a ...
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Brauer equivalence classes over $\mathbb{Q}$.

How to show that there are infinitely many Brauer equivalence classes over $\mathbb{Q}$? I proved that every such class has exactly one division algebra in it, so the question reduces to showing that ...
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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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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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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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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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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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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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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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A simple Artinian left quotient ring of a left Noetherian domain

Recall the following known result, Theorem 6.1: A ring $R$ is a prime left Goldie ring if and only if $R$ has a left quotient ring which is a matrix ring over a division ring (= simple Artinian). Now,...
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Decomposing $\mathbb{F}_p[G]$ ($G$ finite) into products of matrix rings over fields

I have recently begun learning about group algebras over finite fields but am still a little uncertain about these guys. So I was looking for some clarification and verification. Consider the ...
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Unwind quaternion multiplication

I am trying to understand quaterions division. Imagine I have the following equation, where every member is a quaternion: $$Q = (qq_1)(qq_2)...(qq_n)$$ I suppose that, if I maintain the order of ...
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Involutions on endomorphisms over division rings

Let $D$ be a division ring, and let $M $ be a free left $D$-module of finite rank. Assume that $x\mapsto x^*$ is an involution on the ring $\operatorname{End}_D(M)$ (which in this case means: ${}^*$ ...
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Division algebra over 2-adic fields

Let $D$ be the quaternion division algebra and $O$ be a maximal $\mathbb{Z}$-order in $D$, say the Hurwitz quaternion integers. It can be proved that $D$ and $O$ split at odd primes, that is $$D\...
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Has the Riemann Hypothesis been generalized to the Octonions and the Quaternions?

I've noticed that it uses imaginary numbers. I know that sometimes when I have too few dimensions like (-1)^n, dots show where I might expect lines due to imaginary numbers. So perhaps there is a ...
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A corollary of Niven

Please proof corollary of Niven: For $a \in D\backslash R$, the equation ${t^n} = a$ has exactly $n$ solutions in $D$, all of which lie in $R\left( a \right)$, in there $R$ is a real-closed field and $...
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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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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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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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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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34 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 $\...
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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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Dimension of division rings extension

Let $A \subseteq B$ be two division $k$-algebras, where $k$ is a field of characteristic zero. I am not sure if I wish to further assume that $B$ is affine over $A$, namely, if $B$ is finitely ...
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The ring of quotients of the first Weyl algebra

Since there are no comments to this question, I now restrict it to the following question: It is known that: A ring $R$ is a prime left Goldie ring if and only if $R$ has a left quotient ring which ...
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Cannot get the correct result following the division algorithm

I have the following algorithm which is supposed to be a Division algorithm: where m is a's number of digits and n is the b's number of digits. I tried to follow the steps taking a = 42 , b = 6 and ...
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concept of conjugacy class in a ring

Can we think of a similar concept of a conjugacy class in a ring which satisfies two three properties like conjugacy classes. I think of a set $M_x={xyx^{-1}-y}$ for $x\in R$ and $R$ is a division ...
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60 views

Cyclic division ring

Suppose that $D$ is a division ring with center $F$ and with index $p$ prove that $D$ is cyclic if and only if there exists $x$ $\notin$$F$ which $x$$^p$ $\in$$F$.
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A corollary to the Wedderburn-Artin theorem.

Suppose we proved the Wedderburn-Artin theorem, i.e. we have the fact that if S is a semisimple algebra over a field $F$, then $$ A \cong M_{n_1} (D_1) \times ... \times M_{n_k} (D_k), $$ where $D_1,.....
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Additive commutators and the center of a division ring (2)

Can someone proof the following? If $D$ is a division algebra over a field $F$, then as a $Z(D)$ algebra, $D$ is generated by the additive commutators (elements of the form $xy-yx$) Thanks!