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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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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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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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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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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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 ...
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 ...
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 ...