# 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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### Algebra Isomorphism and embedding between matrix algebra and tensor products of algebras

Let $\mathbb{R}$ denote real numbers. For which $\mathbb{R}$-algebra $A$ are $Mat_n(\mathbb{R} \otimes A)$ and $Mat_n(\mathbb{R}) \otimes A$ algebra isomorphic? If $A$ is finite dimensional with ...
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### Wedderburn Artin's theorem for algebras over a field

I learned about the Wedderburn Artin's theorem for simple left artinian ring, says that if $R$ is simple left Artinian ring then $R\cong\mathrm{M}_n(\Delta)$, for some division ring $\Delta$. I want ...
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### Group representation classification via Frobenius theorem

I am struggling to understand the following argument on the classification of irreducible representation as real, complex or quaternionic: Let $G$ be a group and $V$ a vector space over $\mathbb{C}$. ...
1 vote
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### Sub division rings of dimension 2 of division rings

Suppose $A$ is a division ring and $B$ is a sub division ring such that $A$, as a left vector space over $B$, has dimension $2$. Is it true that $B$ must be commutative ?
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### Dimension of division ring over a sub division ring

Let $L$ be a division ring ("skew field") and $K$ a sub division ring. Now suppose that $L$, as a left vector space over $K$, has finite dimension $m$. Does $L$, as a right vector space over ...
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### Is there an Archimedean Dedekind-complete ordered division ring of characteristic zero that is non-isomorphic to the real line?

I know an Archimedean Dedekind-complete ordered field of characteristic $0$ must be isomorphic to $\mathbb{R}$. My question is what if I start from an ordered division ring? Would it still be ...
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### Can we find a normal nilpotent subgroup $G$ of monomial subgroup of $GL_n(D)$ such that $F[G]=M_n(D)$?

Let $D$ be a non-commutative division ring of finite dimension over its center $F$. Also, $n>1$ is a natural number. Consider that $M$ be the monomial subgroup of $GL_n(D)$(containing $n \times n$ ...
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### Continuous addition and multiplication on Euclidean space (dimension > 2) making it into a field?

While TAing a linear algebra class, I happened upon the following question: Question: for $n\geq 3$, are there continuous operations $+, \cdot : \mathbb R^n \times \mathbb R^n \to \mathbb R^n$ that ...
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### Is the spectrum of an element in an algebra, $\sigma(x) = \{z \in\mathbb{C}: \vert z\vert\leq\Vert x \Vert\}$? [closed]

In the book, a first course in functional analysis by D.Somasundaram, it is mentioned that $\sigma(x) = \{z \in\mathbb{C}: \vert z\vert\leq\Vert x \Vert\}$ But the proof is given only for one side ...
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### The multiplicative group of the real quaternion division ring

Let $\mathbb{H}$ be the real quaternion division ring, that is, $\mathbb{H}$ consists of all elements of the form: $a+bi+cj+dk$ in which $a,b,c,d\in\mathbb{R}$ and $i^2=j^2=k^2=-1,ij=-ji=k$ with usual ...
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### Are there any infinite-dimesional division algebras over the real numbers? [duplicate]

I have read that the only finite-dimesional division algebras over the real numbers have dimensions 1, 2, 4, and 8. Are there any infinite-dimensional division algebras over the real numbers?
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### Are the dimensions of division algebras over the real numbers related to with generalizations of Euler's four square identity?

I know that the only division algebras over the real numbers have dimension $1, 2, 4,$ and $8$ (real numbers, complex numbers, quaternions, octonions). I also know that those are the only numbers of ...
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1 vote
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### If E is a splitting field for A then it is clear that there exists a finitely generated subfield E'/F that is also a splitting field. [closed]

I saw in Jacobson's book Finite Dimensional Division Algebras (1996) page 158 the sentence "If E is a splitting field for A then it is clear that there exists a finitely generated subfield E'/F ...
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### division rings $D,D'$ both of finite dimension over their center $F$ are isomorphic as rings iff isomorphic as $F$-algebras?

Say I have two division rings, $D,D'$, both with center $F$ and both are of finite dimension over $F$, for some field $F$. Now suppose that $D\cong D'$ as rings, does if follow that $D\cong D'$ as $F$ ...
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### A consequence of Hasse's reciprocity law of simple algebras

I read somewhere in Godement-Jacquet's book (Zeta functions of simple algebras) the following claim: Let $D$ be a division algebra over a number field $F$ of dimension $d^2$. For each place $v$ of $F$,...
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### Octonions not an associative division algebra?

On this wikipedia page, I read The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field R of real numbers, which are finite-...
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### Tensoring a skew field does not introduce zero divisors

Let $D$ be a skew field with centre $K$ and maximal subfield $E$. Let $F$ be a finite extension of $K$ disjoint to $E$, that is, $F\cap E=K$. Is it true that the tensor product $D\otimes_K F$ has no ...
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### Brauer group of a subfield

Let $A$ be a central simple algebra with a finite dimension over the field $F$. Let $A \supset K \supset F$ be a subfield. Show that $C_A(K)$ and $A \otimes_F K$ are both central simple algebras over ...
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### How to proof there is no idempotent element other than 0 and 1 in a Division Algebra?

If $A$ is a division $K$-algebra. Then I need to proof there is no idempotent element other than $0$ and $1_A$ in $A$. I tried this way : If $0,1_A\neq a\in A$ such that $a^2=a.~$ Now $A$ is division ...
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### Norm of the Sedenions

Let the Cayley-Dickson doubling of the octonions be called the sedenions. The sedenions are not a division algebra, because they contain zero divisors. The presence of zero divisors means that the ...
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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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