# 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.

157 questions
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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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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 $C(C(A)... 1answer 116 views ### Name of Octonions With Biquaternion Coefficients? Ordinary biquaternions are quaternions$(\mathbb{H})$whose coefficients are complex$(\mathbb{C})$. What is the name, analogous to "biquaternions", for octonions$(\mathbb{O})$whose coefficients ... 0answers 137 views ### Do Hopf bundles give all relations between these “composition factors”? Write a fiber bundle$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$, but ... 1answer 162 views ### Hasse invariants under extension of scalars Let$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)$then maps injectively into the ... 0answers 143 views ### A Condition for division ring being a field. I'm reading a paper of Maurice Chacron and I.N. Herstein entitled Powers of skew and symmetric elements in division rings. At the first page of the paper, I got stuck in a problem that: "if in a ... 1answer 58 views ### Ring-theoretic characterization of scalar matrices over a division ring Let$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)$is the subring of scalar matrices, is it generally true that every ring automorphism ... 0answers 33 views ### Decomposing$\mathbb{F}_p[G]$($G$finite) into products of matrix rings over fields I have recently begun learning about group algebras over finite fields but am still a little uncertain about these guys. So I was looking for some clarification and verification. Consider the ... 1answer 135 views ### Prove that the following four statements are equivalent. Prove that the following statements are equivalent for a nonzero ring D: (i)$D$is a division ring. (ii) For all$ a, b ∈ D $with$ a \neq 0 $, the equations$ ax = b $and$ ya = b $have unique ... 0answers 51 views ### Cannot get the correct result following the division algorithm I have the following algorithm which is supposed to be a Division algorithm: where m is a's number of digits and n is the b's number of digits. I tried to follow the steps taking a = 42 , b = 6 and ... 2answers 294 views ### Division algebra = Field? Is a division algebra a field? If not, why does it differ? It is an abelian group with multiplication and division. How not a field? 2answers 98 views ### An exchange property for bases of a free module over a division ring$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$. Here's ... 1answer 258 views ### Dimension of maximum subfield. Let$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$... 0answers 75 views ### 4-D lattices and quaternions It is easy to prove that there are only 2 extensions$\mathbb{Q}(a)$, with$|a|=1$, of$\mathbb{Q}$where$\mathbb{Z}[a]$becomes a lattice(discrete free abelian subgroup of rank 2) in the complex ... 0answers 92 views ### Existence of a division ring on a field. Suppose that$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$if and ... 0answers 55 views ### concept of conjugacy class in a ring Can we think of a similar concept of a conjugacy class in a ring which satisfies two three properties like conjugacy classes. I think of a set$M_x={xyx^{-1}-y}$for$x\in R$and$R$is a division ... 1answer 167 views ### Parallelization of a Sphere gives Division Algebra Is there an elementary proof of the fact, that a parallelization of$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)$denote ... 1answer 159 views ### Brauer group of cyclic extension of the rationals I am trying to compute the relative Brauer group of the cyclic Galois extension$L=\mathbb Q[x]/(x^3-3x+1)$of$\mathbb Q$. I know that $$\mathrm{Br}(L/\mathbb Q)\cong H^2(G,L^*)\cong\mathbb Q^*/N(L^*... 2answers 244 views ### Why do division algebras always have a number of dimensions which is a power of 2? Why do number systems always have a number of dimensions which is a power of 2? Real numbers: 2^0 = 1 dimension. Complex numbers: 2^1 = 2 dimensions. Quaternions: 2^2 = 4 dimensions. ... 2answers 316 views ### Confusion about division in Clifford Algebra On page 202 of The Road to Reality, Penrose claims that if we want to generalize Quaternions to n dimensions using Clifford Algebra, we must abandon the division property. I have a hard time believing ... 0answers 57 views ### Is the generalized hopf map of an alternative finite-dimensional real division algebra continuous? Let A be an n-dimensional alternative real division algebra (not necessarily associative). Is the map$$ \eta \colon \bigl\{(x,y) \in A \times A : |x|^2+|y|^2=1\bigr\} \to A \sqcup \{\infty\}, \... 2answers 601 views ### 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. (... 0answers 126 views ### 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,... 0answers 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$$F. 2answers 174 views ### Central division algebras and splitting fields Let K be a field and D be a central division algebra over K of degree n. Suppose that L\subset D is a maximal subfield, so that [L:K]=n. Then we know that L is a splitting field, so ... 2answers 431 views ### Minimal projections on von Neumann Algebras A projection p \neq 0 in a von Neumann Algebra A is called minimal, if for every projection 0\neq q\in A with q \leq p already q=p. I want to prove the following theorem: For a minimal ... 2answers 240 views ### Field extension whose tensor product with itself over \mathbb{Q} is not a field An old qual problem reads Let D be a 9-dimensional central division algebra over \mathbb{Q} and K \subset D be a field extension of \mathbb{Q} of degree >1. Show that K \otimes_\... 0answers 133 views ### Motivating the Cayley-Dickson construction by proving Hurwitz's theorem To me it seems the way to motivate the Cayley-Dickson construction is to prove Hurwitz's theorem, which is done over at Wikipedia. The theorem states the only real division algebras equipped with a ... 1answer 75 views ### Polarization identity 2(a,b)(c,d)=(ac,bd)+(ad,bc) I am interested in following along this Wikipedia article's derivation of properties of composition algebras (in particular, Euclidean Hurwitz algebras). Let A be a unital, not necessarily ... 1answer 117 views ### Version of Wedderburn's theorem on central simple algebras Suppose that A be a central simple algebra over a field k. Then by Wedderburn's theorem A\cong M_n(D) for some division k-algbera D. But to define the 'Brauer equivalence' I need that D is ... 0answers 103 views ### Unwind quaternion multiplication I am trying to understand quaterions division. Imagine I have the following equation, where every member is a quaternion:$$Q = (qq_1)(qq_2)...(qq_n)$$I suppose that, if I maintain the order of ... 2answers 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 ... 1answer 116 views ### 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 ... 1answer 244 views ### 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 ... 1answer 62 views ### 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 ... 1answer 158 views ### 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\} ... 3answers 378 views ### 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)? 0answers 64 views ### 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: {}^* ... 1answer 215 views ### 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 ... 2answers 77 views ### 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?... 0answers 43 views ### 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\...
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) + (...