scholarly journals Splitting of Algebras by Function Fields of One Variable

1966 ◽  
Vol 27 (2) ◽  
pp. 625-642 ◽  
Author(s):  
Peter Roquette
Keyword(s):  
Tensor Product ◽  
Function Fields ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Field Extension ◽  
Brauer Group ◽  
Central Simple Algebras ◽  
Natural Mapping ◽  
Simple Algebras ◽  
Central Simple

Let K be a field and (K) the Brauer group of K. It consists of the similarity classes of finite central simple algebras over K. For any field extension F/K there is a natural mapping (K) → (F) which is obtained by assigning to each central simple algebra A/K the tensor product which is a central simple algebra over F. The kernel of this map is the relative Brauer group (F/K), consisting of those A ∈(K) which are split by F.

UNIT GROUPS OF CENTRAL SIMPLE ALGEBRAS AND THEIR FRATTINI SUBGROUPS

2010 ◽  
Vol 09 (06) ◽  
pp. 921-932 ◽  
Author(s):  
R. FALLAH-MOGHADDAM ◽  
M. MAHDAVI-HEZAVEHI
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Global Field ◽  
Central Simple Algebras ◽  
Frattini Subgroup ◽  
Division Rings ◽  
Finite Dimensional ◽  
Simple Algebras ◽  
Central Simple

Given a finite dimensional F-central simple algebra A = Mn(D), the connection between the Frattini subgroup Φ(A*) and Φ(F*) via Z(A'), the center of the derived group of A*, is investigated. Setting G = F* ∩ Φ(A*), it is shown that [Formula: see text] where the intersection is taken over primes p not dividing the degree of A. Furthermore, when F is a local or global field, the group G is completely determined. Using the above connection, Φ(A*) is also calculated for some particular division rings D.

FRATTINI SUBGROUP OF THE UNIT GROUP OF CENTRAL SIMPLE ALGEBRAS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250061
Author(s):  
H. R. DORBIDI ◽  
M. MAHDAVI-HEZAVEHI
Keyword(s):  
Central Simple Algebra ◽  
Rational Functions ◽  
Simple Algebra ◽  
Unit Group ◽  
Closed Field ◽  
Central Simple Algebras ◽  
Frattini Subgroup ◽  
Group A ◽  
Simple Algebras ◽  
Central Simple

Given an F-central simple algebra A = Mn(D), denote by A′ the derived group of its unit group A*. Here, the Frattini subgroup Φ(A*) of A* for various fields F is investigated. For global fields, it is proved that when F is a real global field, then Φ(A*) = Φ(F*)Z(A′), otherwise Φ(A*) = ⋂p∤ deg (A) F*p. Furthermore, it is also shown that Φ(A*) = k* whenever F is either a field of rational functions over a divisible field k or a finitely generated extension of an algebraically closed field k.

On the Brauer Group of Algebras Having a Grading and an Action

1976 ◽  
Vol 28 (3) ◽  
pp. 533-552 ◽  
Author(s):  
Morris Orzech
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Bilinear Form ◽  
Brauer Group ◽  
Central Simple Algebras ◽  
Simple Algebras ◽  
Central Simple

Beginning with Wall's introduction [19] of Z2-graded central simple algebras over a field K, a number of related generalizations of the Brauer group have been proposed. In [16] the field K was replaced by a commutative ring R, building upon the theory developed in [1]. The concept of a G-graded central simple K-algebra (G an abelian group) was first defined in [12]; this work and that of [16] was subsequently unified in [6] and [7] via the construction and computation of the graded Brauer group Bφ﹛R, G) (φ a bilinear form from G X G to U(R), the units of R).

scholarly journals Local–global principle for reduced norms over function fields of -adic curves

2017 ◽  
Vol 154 (2) ◽  
pp. 410-458 ◽  
Author(s):  
R. Parimala ◽  
R. Preeti ◽  
V. Suresh
Keyword(s):  
Local Field ◽  
Function Field ◽  
Function Fields ◽  
Residue Field ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Hasse Principle ◽  
Central Simple ◽  
Global Principle ◽  
Reduced Norm

Let $K$ be a (non-archimedean) local field and let $F$ be the function field of a curve over $K$. Let $D$ be a central simple algebra over $F$ of period $n$ and $\unicode[STIX]{x1D706}\in F^{\ast }$. We show that if $n$ is coprime to the characteristic of the residue field of $K$ and $D\cdot (\unicode[STIX]{x1D706})=0$ in $H^{3}(F,\unicode[STIX]{x1D707}_{n}^{\otimes 2})$, then $\unicode[STIX]{x1D706}$ is a reduced norm from $D$. This leads to a Hasse principle for the group $\operatorname{SL}_{1}(D)$, namely, an element $\unicode[STIX]{x1D706}\in F^{\ast }$ is a reduced norm from $D$ if and only if it is a reduced norm locally at all discrete valuations of $F$.

scholarly journals Local selectivity of orders in central simple algebras

2017 ◽  
Vol 13 (04) ◽  
pp. 853-884 ◽  
Author(s):  
Benjamin Linowitz ◽  
Thomas R. Shemanske
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Ring Of Integers ◽  
Isomorphism Classes ◽  
Maximal Orders ◽  
Local Variant ◽  
Simple Algebras ◽  
Central Simple ◽  
Global Principle

Let [Formula: see text] be a central simple algebra of degree [Formula: see text] over a number field [Formula: see text], and [Formula: see text] be a strictly maximal subfield. We say that the ring of integers [Formula: see text] is selective if there exists an isomorphism class of maximal orders in [Formula: see text] no element of which contains [Formula: see text]. In the present work, we consider a local variant of the selectivity problem and applications. We first prove a theorem characterizing which maximal orders in a local central simple algebra contain the global ring of integers [Formula: see text] by leveraging the theory of affine buildings for [Formula: see text] where [Formula: see text] is a local central division algebra. Then as an application, we use the local result and a local–global principle to show how to compute a set of representatives of the isomorphism classes of maximal orders in [Formula: see text], and distinguish those which are guaranteed to contain [Formula: see text]. Having such a set of representatives allows both algebraic and geometric applications. As an algebraic application, we recover a global selectivity result, and give examples which clarify the interesting role of partial ramification in the algebra.

scholarly journals Derivations in central separable algebras

1978 ◽  
Vol 19 (1) ◽  
pp. 75-77 ◽  
Author(s):  
George T. Georgantas
Keyword(s):  
Galois Group ◽  
Brauer Group ◽  
Normal Extension ◽  
Noether Theorem ◽  
Central Simple Algebras ◽  
Group Extensions ◽  
Simple Algebras ◽  
Central Simple ◽  
Inner Automorphisms ◽  
Separable Algebras

Given N a finite separable normal extension of a field F, it is well known that the Brauer group Br(N/F) of classes of central simple F-algebras split by N is isomorphic with Ext(N*, G), the classes of group extensions of N* by the Galois group G of N over F. In the construction of this isomorphism, a key role is played by the Skolem-Noether Theorem which extends automorphisms to inner automorphisms in central simple algebras.

scholarly journals More on the Schur group of a commutative ring

1985 ◽  
Vol 8 (2) ◽  
pp. 275-282 ◽  
Author(s):  
R. A. Mollin
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Homomorphic Image ◽  
Group Ring ◽  
Local Field ◽  
Natural Generalization ◽  
Brauer Group ◽  
Central Simple Algebras ◽  
Simple Algebras ◽  
Central Simple

The Schur group of a commutative ring,R, with identity consists of all classes in the Brauer group ofRwhich contain a homomorphic image of a group ringRGfor some finite groupG. It is the purpose of this article to continue an investigation of this group which was introduced in earlier work as a natural generalization of the Schur group of a field. We generalize certain facts pertaining to the latter, among which are results on extensions of automorphisms and decomposition of central simple algebras into a product of cyclics. Finally we introduce the Schur exponent of a ring which equals the well-known Schur index in the global or local field case.

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