# Questions tagged [central-simple-algebras]

The tag has no usage guidance.

54 questions
Filter by
Sorted by
Tagged with
1 vote
70 views

### A possible generalization of Brauer's theorem about the prime factors of the period and index of a central simple algebra

Let $K$ be an arbitrary field, and let $K^s$ be a fixed separable closure of $K$. Let $F/K$ a be a finite Galois extension in $K^s$. Let $n>0$ be a natural number. Let $A$ be a central simple ...
• 13.8k
1 vote
342 views

### Amitsur's theorem for generalized Severi–Brauer varieties

Let $k$ be a field of characteristic zero and assume that $A$ is a central simple algebra of index $2^n > 2$. We denote by $\operatorname{SB}_i(A)$ the $i$-th (generalized) Severi–Brauer variety of ...
• 1,419
61 views

### Anisotropic kernel of groups of type A

I'm studying the results of classification of reductive groups using Tits index and anisotropic kernel. It is known that simple groups with Tits index $^1 A_{n,r}^{(d)}$ are of the form $SL_{r+1}(D)$, ...
• 321
232 views

• 121
814 views

### Brauer group of rational numbers

Reading about the calculation of the Brauer group of rational numbers, the calculations of the group are extremely lengthy and technical. First of all, it will be very helpful to me if someone can ...
295 views

• 13.8k
1 vote
61 views

### Maximal orders separable over their centre

Let $\mathcal{A}$ be a central simple $K$-algebra, where $K$ is an algebraic number field. It is known that $\mathcal{A}$ is separable over $K$ (following the definition of DeMeyer and Ingraham's book)...
• 323
240 views

• 262
316 views

### Azumaya algebra and central separable algebra over local rings

let be $R$ a commutative ring Auslander and Goldman define a central separabl $R$-algebra $A$ as a faithful $R$-module with $Z(A) = R$ , and such that $A$ is a projective $A\otimes_R A^{\mathrm{op}}$-...
69 views

• 173
656 views

### Jacobson radical of a tensor product

Let $R$ be a commutative ring and $A_1$, $A_2$ be $R$-algebras. Is there any general mean to compute the Jacobson radical of the tensor product $A_1\otimes_R A_2$ in terms of ${\rm Rad}(A_i )$, $i=1,2$...
• 6,259
168 views

### Crossed product division algebra

Let $k$ be a field, $D$ be a crossed product division algebra over $k$, namely $D$ has a maximal subfield which is Galois over $k$. Is it possible $D$ contain some other maximal subfield which is non-...
491 views

### Elementary proof that a central simple algebra over a field having a maximal subfield is a cyclic algebra

I'm currently reading the book "Central Simple Algebras and Galois Cohomology" written by Philippe Gille and Tamas Szamuely. In the book, I don't understand a computational proof of the theorem that ...
• 1,053
67 views

• 103
345 views

### Brauer groups and field extensions

Let $k$ be a field and $\mathrm{Br}(k)$ the Brauer group of $k$. Let $k \subset L$ be a field extension. Let $b \in \mathrm{Br}(k)$ and denote by $b \otimes L \in \mathrm{Br}(L)$ the base-change of $b$...
• 20.9k
313 views

### Interaction between the Brauer group and abelian extensions

If $k$ is a field of characteristic zero and $k$ has no (non-trivial) abelian extensions (e.g. the composite of all solvable extensions of $\mathbb{Q}$), then $\text{Br(k)} = 0$ by the norm residue ...
• 209
535 views

• 5,033
1k views

### Non-commutative Galois theory

Recall that an finite-dimensional algebra $A$ over a field $k$ is central simple iff there is an iso $A \otimes_k A^{op} \cong M_n(k)$ where $A^{op}$ is the opposite ring and $M_n(k)$ is the matrix ...
• 1,986
361 views

### When is a crossed-product algebra a division algebra?

Let $L/K$ be a finite Galois extension with Galois group $G$. For every 2-cocycle $\gamma$ of $G$ with values in $L^\times$ there is the crossed-product $K$-algebra S(L,G,\gamma) = \bigoplus_{g\in ...
1 vote
211 views

### Dimension of the center of a subalgebra of a simple algebra

Let $F$ be a field. Let $A$ be a simple (associative unital) $F$-algebra with center reduced to $F$. Let $B$ be a $F$-subalgebra of $A$; assume that $A$ is can be generated as left $B$-module by $n$ ...
• 11
324 views

I am currently working on level issues in the division central simple algebra case, say $D$ over a local non-archimedean field $F$ (e.g. $\mathbf{Q}_p$). Let say that $\mathcal{O}_D$ and $\mathcal{O}... • 5,344 9 votes 1 answer 597 views ### Orders in Central Simple Algebras. Applications It is known that orders in quaternion algebras (over a number field) are used for constructing geometric objects like hyperbolic orbifolds and Shimura curves. Moreover, if one knows embedding ... • 93 4 votes 1 answer 179 views ### Let$A\in\operatorname{M}_n(F)$be a matrix, how to prove$\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$I asked the following question on Math Stack Exchange, but no people reply. I know MO is more professional and it is for mathematicians to discuss research problems. Maybe this question is unsuitable ... • 243 4 votes 1 answer 174 views ### Does every equivalence class in a Brauer-Wall group have a (graded) division algebra? It is known that each equivalence class in a Brauer group has a division algebra (or, in other words, every central simple algebra is isomorphic to$\mathrm{Mat}(D)$for some division algebra$D$). Is ... • 305 6 votes 1 answer 261 views ### Infinite dimensional simple algebras of finite degree Let$F$be a field and let$B$be an$F$-algebra. The degree of$B$over$F$is the smallest positive integer$\deg_F B = d \geq 1$such that every element of$B$satisfies a (monic) polynomial of ... • 2,949 4 votes 1 answer 827 views ### Regular or elliptic elements in the multiplicative group of central division algebra For an element$g$of a connected reductive group$G$over a field$F$,$g$is called$regular$if the dimension of the centralizer of$g$is equal to the rank of the algebraic group$G$,$g$is ... • 945 3 votes 0 answers 340 views ### Discussion of specific arithmetic triangle groups? Arithmetic triangle groups were classified in Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan Volume 29, Number 1 (1977), 91-106. The (2,3,7) case was discussed in detail in a number of ... • 15.7k 10 votes 0 answers 725 views ### What does the tensor product of two central simple algebras correspond to geometrically? Let$k$be a field, assumed to have characteristic$0$for simplicity (though this probably isn't necessary). Let$A$be a central simple algebra over$k$of dimension$n^2$. Then the collection of ... • 20.9k 15 votes 3 answers 1k views ### How to distinguish division algebras from matrix algebras? Suppose$D$is an explicitly given rank 9 central simple algebra over${\mathbb Q}$(or a number field). For example it could be specified by two cubic subfields${\mathbb Q}(a), {\mathbb Q}(b)\subset ...
• 5,293
Let $k$ be a field which I don't suppose to be algebraically closed. Then, the endomorphism ring of any irreducible representation of a finite group over $k$ is a division ring. What is known about ...