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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 ...
Mikhail Borovoi's user avatar
1 vote
0 answers
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 ...
nxir's user avatar
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3 votes
0 answers
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 $SL_{r+1}(D)$, ...
YJ Kim's user avatar
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2 votes
1 answer
232 views

On Dirac/ Clifford matrices

Let $(\eta^{\mu\nu})=\operatorname{diag}(+1,-1,-1,-1)$. The Dirac matrices $\gamma^\mu$, $\mu=0,1,2,3$ satisfy by definition $$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\tag{1}\label{1}$$ where $\{A,B\}=...
asv's user avatar
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3 votes
1 answer
277 views

Unitary involutions on a simple central algebra after a scalar extension

$\DeclareMathOperator{id}{id}$ Let $L/K$ be a quadratic separable extension of fields. Let $A$ be a central simple algebra over $L$ such that its norm $N_{L/K}(A)$ splits. Then we know that there ...
Haowen Zhang's user avatar
5 votes
0 answers
207 views

Essence of relations between central simple algebras and Galois cohomology in canonical morphism of class field theory

I was somewhat puzzled after I finished learning class field theory for several times. My question is about the relations between "classical simple algebras, Brauer groups" and "modern ...
youknowwho's user avatar
2 votes
0 answers
84 views

What is the relationship between ramification in central simple algebras and in fields?

Suppose $K$ is the field of fractions of a Dedekind domain $R$, and let $L$ be a finite extension of $K$. There is a notion of ramification of primes of $K$ in $L$, which describes how $\mathfrak p \...
user's user avatar
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6 votes
1 answer
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 ...
Siavosh Ossareh's user avatar
9 votes
2 answers
295 views

What is the quotient group $D^*/{F^*(1+P_D)}$ for a quaternion division algebra $D$ over a local field $F$?

Let $F$ be the non-archimedean local field $\mathbb{Q}_p$ for some prime $p$ and $D$ be a quaternion division algebra over $F$. Let $\mathcal{O}_D$ and $\mathcal{P}_D$ denote the ring of integers of $...
BPK's user avatar
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Are there any central simple algebras admitting a standard basis?

Are there any central simple algebras admitting a standard basis? By a standard basis I mean a normal basis that has a cyclic property generalizing that of the familiar basis $1, i, j, k$ for ...
Andy's user avatar
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51 views

Subalgebra of a crossed product central division algebra, generated by powers of group elements

Let $k=\mathbb C$, let $K$ be a finite extension of the field $k(X_1,X_2,X_3)$ of rational functions in $3$ variables, let $L/K$ be a finite Galois extension of commutative fields and let the Galois ...
Łukasz Grabowski's user avatar
3 votes
0 answers
201 views

The Brauer group and the second Galois cohomology group

I know that for any finite Galois extension K of $\mathbb{Q}$ the group $H^2(Gal(K/\mathbb{Q}),K^*)$ is isomorphic to the Brauer group $Br(K/\mathbb{Q})$. The isomorphism goes as follows: to a 2-...
Jacques's user avatar
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Multiplication law in a central simple algebra of dimension 9 over a global field

Let $k$ be a global field, for example $k=\Bbb Q$. Let $D$ denote the central simple algebra of dimension 9 over $k$ with given local invariants $i_v$. Here $v$ runs over the set $\Omega_f(k)$ of ...
Mikhail Borovoi's user avatar
2 votes
0 answers
74 views

Equivalence classes of Hermitian elements in a central simple algebra with an involution of second kind

Let $k_0$ be a field of characteristic 0, $k/k_0$ be a quadratic extension, and $A/k$ be a central simple algebra over $k$ of dimension $9=3^2$ with an involution of second kind $\sigma$. Then $^\...
Mikhail Borovoi's user avatar
1 vote
0 answers
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)...
a196884's user avatar
  • 323
6 votes
1 answer
240 views

Hasse invariant and the Clifford algbera

Let $$q = a_1 x_1^2 + \cdots + a_n x_n^2$$ be a quadratic form over some $p$-adic field $\mathbb{Q}_p$. We thus have its Hasse invariant $$\mathcal{h}(q) = \prod_{1 \leq i < j \leq n} (a_i,a_j)_p \...
Rita's user avatar
  • 103
4 votes
0 answers
223 views

Fixing error in a proof from "Central simple algebras and Galois cohomology"

I'm trying to understand the proof of Proposition 2.2.10 in Gille-Samuely's book "Central simple algebras and Galois cohomology" (2nd edition), and I believe it has an error. Here's the ...
Rita's user avatar
  • 103
3 votes
0 answers
79 views

The type number of an algebra

I've been reading On the existence of maximal orders, by C.F. Yu, in which he discusses maximal $R$-orders in semisimple algebras over a field $K$, where $R$ is a Noetherian integral domain and $K = \...
a196884's user avatar
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3 votes
0 answers
142 views

Central simple algebras via cohomology

I am following the book Central Simple Algebras and Galois Cohomology, by Gille and Szamuely (I am using the second edition). In section 2.4, the authors remark that the tensor product induces a ...
user50139's user avatar
  • 525
3 votes
1 answer
263 views

Elementary classification of division rings

Are there examples (other than the two mentioned below) of fields $K$ such that the classification of all finite dimensional division $K$-algebras is possible using only elementary theory (lets say a ...
Mare's user avatar
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3 votes
0 answers
162 views

Multiplicative structure of the K-theory of Severi-Brauer varieties

There is a well-known result by Quillen stating that if $X_A$ is the Severi-Brauer variety of a central simple algebra $A$ of degree $d$ over a field $k$, then its (Quillen) K-theory decomposes as $$...
Captain Lama's user avatar
3 votes
0 answers
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 $A\otimes_R A^{\mathrm{op}}$-...
Moutand Mohammed's user avatar
3 votes
0 answers
69 views

Splitting of central simple algebras in the Schur subgroup over residue fields of places

​Recall that a valuation domain of ​a field extension ​$K/k$ is a $k$-subalgebra $V$ of $K$ ​not equal to $K$ ​such that for every $a\in K$ at least one of $a$ and $a^{-1}$ is in $V$. A​ ​place of $K/...
Fabian Meumertzheim's user avatar
6 votes
1 answer
735 views

Example of a central simple algebra

Let $A$ be a finite dimensional central simple algebra over a field $F$ of characteristic $0$. So by Weddernburn's theorem, $A\cong M_n(D)$ for some division algebra $D$ over $F$. Let $\dim_F(D)=m^2$....
Soluble's user avatar
  • 1,149
4 votes
1 answer
320 views

Cyclotomic fields and splitting of central simple algebras

Let $K$ be a cyclotomic field of degree $n$ and $A$ a central simple algebra over $\mathbb{Q}$ of dimension $n^2$. How can one determine whether there is a $\mathbb{Q}$-algebra embedding $K \...
Sun Ra's user avatar
  • 173
6 votes
0 answers
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 ${\rm Rad}(A_i )$, $i=1,2$...
Paul Broussous's user avatar
2 votes
2 answers
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-...
user avatar
4 votes
1 answer
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 ...
gualterio's user avatar
  • 1,053
3 votes
0 answers
67 views

Spliting of division algebras over a ring

Let $D$ be a central division algebra over $\mathbb{Q}_p$, of dimension $n^2$. For example, let $D=E_n(\Pi)$, where $E_n$ is the unramified degree extension of $\mathbb{Q}_p$ such that $\Pi^n=p$ and $\...
wuzx's user avatar
  • 517
3 votes
1 answer
78 views

Splitting variety for bicyclic algebras

Let $F$ be a field contains a primitive root of unity of order $p$, where $p$ is a prime number. Let $a,b \in F^\times$, then one can look at the cyclic algebra $(a,b)_p \in {_p}Br(F)$ where ${_p}Br(F)...
Guy's user avatar
  • 103
6 votes
1 answer
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 $b$...
Daniel Loughran's user avatar
3 votes
0 answers
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 ...
Gnuk's user avatar
  • 209
5 votes
3 answers
535 views

Reference request: correspondence between central simple algebras and quadratic forms

Let $A$ be an algebra over $k$, $\operatorname{tr_A}(x, y):=\operatorname{tr}(m_{xy})$ be a trace form on $A$, and $V_A$ be its restriction on the orthogonal complement to $1$. I wonder why a map $A \...
evgeny's user avatar
  • 1,990
4 votes
0 answers
326 views

Is a central simple algebra necessarily cyclic if it splits after a cyclic Galois extension?

Let $A$ be a central simple algebra of degree $n$ over $k$, $\dim_kA=n^2$, let $K/k$ be a cyclic galois extension of degree $n$. Suppose $A\times_kK\cong M_n(K)$, does this imply that $A$ is a cyclic ...
user avatar
2 votes
1 answer
394 views

Dimension of maximal tori in division algebras

Does there exist a field $K$ and a finite-dimensional $K$-division algebra $D$ possessing two maximal separable subfields of different dimensions? Remark: If $D$ is separable ($Z(D)$ a separable ...
Sven Wirsing's user avatar
1 vote
1 answer
333 views

Constructing groups of Type E^{66}_{7,1} having non trivial Tits algebra

This can be considered as a continuation of my last useful question: Constructing groups of Type E7 with certain Tits Index It is known that a quadratic form $q$ of dimension $12$, having splitting ...
nxir's user avatar
  • 1,419
4 votes
0 answers
399 views

Reference for classification of positive involutions

An involution on a finite dimensional algebra $A$ over $\mathbb{Q}$ is an involutive anti-automorphism of $A$. If $\sigma$ is an involution on $A$, we say that $\sigma$ is positive if $\mathrm{Tr}_{A/\...
Aurel's user avatar
  • 5,033
17 votes
1 answer
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 ...
Jakob's user avatar
  • 1,986
4 votes
0 answers
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 ...
Steffen Kionke's user avatar
1 vote
0 answers
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 $n$ ...
tom's user avatar
  • 11
3 votes
1 answer
324 views

On conductors, levels and traces on quaternion algebras

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}...
Desiderius Severus's user avatar
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 ...
Patricio's user avatar
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 ...
HGF's user avatar
  • 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 ...
user50311's user avatar
  • 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 ...
John Voight's user avatar
  • 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 ...
Hiro's user avatar
  • 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 ...
Mikhail Katz's user avatar
  • 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 ...
Daniel Loughran's user avatar
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 ...
Tim Dokchitser's user avatar
4 votes
2 answers
1k views

Galois theory of endomorphism rings of irreducible representations

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 ...
darij grinberg's user avatar