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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.

A division algebra $D$ is a vector space 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 skew field (= 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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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; $D$ is of dimension $9$ over $\mathbb{Q}$ and…
Beginner
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Algebraic division algebra over Euclidean field

By Frobenius' Theorem we know that if $D$ be an algebraic non-commutative division algebra over $\mathbb{R}$ then ,as an $\mathbb{R}$-algebra, $D$ is isomorphic to $\mathbb{H}$. We can also replace $\mathbb{R}$ by any real closed field. if we…
s.Bahari
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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?
Not a grad student
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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 the Galois automorphisms of $K$ over $F$. Then By…
Beginner
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Existence of division algebras with center $\mathbb{Q}$ of prime degree

In First Course in Noncommutative rings of T.Y.Lam (p.210), the author stated that "It is known that for each $n$, there exists a $\mathbb{Q}$-division algebra $A_n$ of dimension $p_n^2$, with $Z(A_n)=\mathbb{Q}$." Here $p_n$ is a prime number and…
user523134
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The dimension of a division ring over its center is square.

Let $D$ be a division ring and let $K$ be the center of $D$. Assume $\dim_K(D)<\infty$. Why is $\dim_K(D)$ a square?
Confused on math
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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 dimension and if $K$ is a maximal subfield of $A$,…
Beginner
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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 prove it? I know we have $A \simeq M_2(F)$ or $A$ is a…
s.Bahari
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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 a ring is when ring is 'center structured' or…
user3520363
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finite integral domain algebras are division algebras

If R is a finite-dimensional algebra over a field k, and if R is also an integral domain, show that R is a division algebra over k.
zübeyir türkoğlu
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$\mathbb{R}$ is closed under division

The set of real numbers $\mathbb{R}$ is closed under division. Does that mean $0$ is also considered? more specifically, should it be $\mathbb{R}-\{ 0 \}$? because division by $0$ is not defined.
rohitt
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Divison algebra vs general/abstract algebra

I am not a mathematician but confused about division algebra vs. algebra. I suspect that "division algebra" is a sub-category(literally, not a math concept) of general or abstract algebra because of following. "A commutative division algebra is…
lockedscope
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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$.
jack
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