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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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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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 proper subring of an 4-dimensional division F-algebra is a field.

I'm stuck with this thing: Let $F$ be a field of characteristic $\neq 2$, let $D$ be a 4 dimensional noncommutative division algebra over $F$. For $x\in D\smallsetminus F, F[x]$ is a field of ...
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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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Is my paper on a number system that allows arithmetic on 3D vectors useful?

I have constructed a number system similar to the quaternions, but with three dimensions, not four, ie vectors of the form $(x, y, z)$. It has fairly well-behaved multiplication and division and every ...
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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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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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Do the octonions form a field?

The octonions are a noncommutative nonassociative normed division algebra over $\mathbb{R}$. Multiplication distributes over addition. Somehow, the existence of a norm implies the existence of ...
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questions about double centralizer theorem

An important fact in the theory of central simple algebras is the double centralizer theorem, which says: if $k$ is a field, $A$ is a $k$-algebra, $V$ is a faithful semisimple $A$-algebra, then $C(C(A)...
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Name of Octonions With Biquaternion Coefficients?

Ordinary biquaternions are quaternions $(\mathbb{H})$ whose coefficients are complex $(\mathbb{C})$. What is the name, analogous to "biquaternions", for octonions $(\mathbb{O})$ whose coefficients ...
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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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Hasse invariants under extension of scalars

Let $K\subset L$ be finite extensions of $\Bbb{Q}$. Background. Let $D$ be a finite dimensional division algebra with center $K$. Its class in the Brauer group $Br(K)$ then maps injectively into the ...
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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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Ring-theoretic characterization of scalar matrices over a division ring

Let $Q$ be a division ring, and let $M_n(Q)$ be the ring of $n \times n$ matrices over $Q$. If $Q' \subset M_n(Q)$ is the subring of scalar matrices, is it generally true that every ring automorphism ...
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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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Prove that the following four statements are equivalent.

Prove that the following statements are equivalent for a nonzero ring D: (i) $D$ is a division ring. (ii) For all $ a, b ∈ D $ with $ a \neq 0 $, the equations $ ax = b $ and $ ya = b $ have unique ...
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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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Division algebra = Field?

Is a division algebra a field? If not, why does it differ? It is an abelian group with multiplication and division. How not a field?
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An exchange property for bases of a free module over a division ring $R$

Let $M$ be a module over a division ring $R$ and let $A$ and $B$ be bases of $M$. Then $\forall a \in A \ \ \exists b \in B: \ \ (A \setminus \{ a \}) \cup \{ b \}$ is a basis of $M$. Here's ...
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258 views

Dimension of maximum subfield.

Let $D$ be a finite central division algebra over $Z$ (the center of $D$), and $F$ be a maximum subfield of $D$ (that is to say, there does not exist a larger subfield $G$ of $D$ such that $G$ ...
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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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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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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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Parallelization of a Sphere gives Division Algebra

Is there an elementary proof of the fact, that a parallelization of $S^n$ can turn $\mathbb{R}^{n+1}$ into a division algebra? My guess was something like this: Let $v_1(x),\dots, v_{n}(x)$ denote ...
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Brauer group of cyclic extension of the rationals

I am trying to compute the relative Brauer group of the cyclic Galois extension $L=\mathbb Q[x]/(x^3-3x+1)$ of $\mathbb Q$. I know that $$ \mathrm{Br}(L/\mathbb Q)\cong H^2(G,L^*)\cong\mathbb Q^*/N(L^*...
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Why do division algebras always have a number of dimensions which is a power of $2$?

Why do number systems always have a number of dimensions which is a power of $2$? Real numbers: $2^0 = 1$ dimension. Complex numbers: $2^1 = 2$ dimensions. Quaternions: $2^2 = 4$ dimensions. ...
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Confusion about division in Clifford Algebra

On page 202 of The Road to Reality, Penrose claims that if we want to generalize Quaternions to n dimensions using Clifford Algebra, we must abandon the division property. I have a hard time believing ...
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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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Finite dimensional division algebra over C

Another abstract algebra question from my university days that has me stumped at where to start! I know what a division ring is and I think I understand what a division algebra over $\mathbb C$ is. (...
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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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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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Central division algebras and splitting fields

Let $K$ be a field and $D$ be a central division algebra over $K$ of degree $n$. Suppose that $L\subset D$ is a maximal subfield, so that $[L:K]=n$. Then we know that $L$ is a splitting field, so ...
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Minimal projections on von Neumann Algebras

A projection $p \neq 0$ in a von Neumann Algebra $A$ is called minimal, if for every projection $0\neq q\in A$ with $q \leq p$ already $q=p$. I want to prove the following theorem: For a minimal ...
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Field extension whose tensor product with itself over $\mathbb{Q}$ is not a field

An old qual problem reads Let $D$ be a 9-dimensional central division algebra over $\mathbb{Q}$ and $K \subset D$ be a field extension of $\mathbb{Q}$ of degree $>1$. Show that $K \otimes_\...
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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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Polarization identity $2(a,b)(c,d)=(ac,bd)+(ad,bc)$

I am interested in following along this Wikipedia article's derivation of properties of composition algebras (in particular, Euclidean Hurwitz algebras). Let $A$ be a unital, not necessarily ...
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Version of Wedderburn's theorem on central simple algebras

Suppose that $A$ be a central simple algebra over a field $k$. Then by Wedderburn's theorem $A\cong M_n(D)$ for some division $k$-algbera $D$. But to define the 'Brauer equivalence' I need that $D$ is ...
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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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240 views

$\Bbb{H}_{\Bbb{Q}}$ is only four dimensional division algebra over rationals.

How to prove that only four dimensional division algebra (noncommutative) over $\Bbb{Q}$ is rational quaternions? After a bit of internet research, I am very sure about the above statement, if not ...
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Show 3D-division algebra over the reals cannot exist using linear algebra

There is a great comment by Jyrki Lahtonen here: Why is quaternion algebra 4d and not 3d? It is not too difficult to show that a 3D-division algebra over the reals cannot exist. If $D$ were such a ...
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Is there any construction of infinite dimensional algebraic division ring?

I know that there is a division algebra over $\mathbb{Q}$ such that it is algebraic and infinite dimensional over it's center i.e. $\mathbb{Q}$. But for construct this division algebra. we can use ...
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Show that a finite dimensional algebra D with identity over a skewfield F is a semifield if and only if it has no zero divisors.

I'm struggling with a proof of the next lemma. Show that a finite dimensional algebra $D$ with identity over a skewfield $F$ is a semifield if and only if it has no zero divisors. EDIT: Actualy I ...
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How to understand that the left regular representation of a division algebra is irreducible?

In Weyl's book The classical groups, it is said the regular representations of a division algebra are faithful and irreducible. The key step is to show the ideal of the division algebra is $\{0\}$ ...
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algebraically closed field in a division ring?

Is it possible to have $K \subset D$ where $K$ is algebraically closed field and $D$ is a division ring such that $K \subseteq Z(D)$?
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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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Finite dimensional central division algebras over a finite extension of $\mathbb{F}_q(T)$

Over number fields, finite dimensional central division algebras are always cyclic algebras. So the construction of cyclic algebras is a nice recipe to create algebras, which exhausts all finite ...
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On Nilpotent Elements of $M_n (F)$

For a field $F$, I have proved that $A \in M_n(F)$ is nilpotent iff $A^n=0$. Now I am curious about Division Rings. If we consider $F$ as a division ring then what happens? Does the result remain true?...
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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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Finding basis and dimension based on definition of space

I've got two vector spaces $U$ and $V$ over division ring $\mathbb{T}$ . Space $W$ over division ring $\mathbb{T}$ is defined as $W =\{( u, v ); u \in U, v \in V \}$ with operations $(u_1, v_1) + (...