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 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.

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 Hermitian forms over odd dimensional algebras

1989 ◽  
Vol 32 (1) ◽  
pp. 139-145
Author(s):  
D. W. Lewis
Keyword(s):  
Galois Group ◽  
Quadratic Forms ◽  
Natural Extension ◽  
Central Simple Algebras ◽  
Witt Group ◽  
Hermitian Forms ◽  
Odd Degree ◽  
Simple Algebras ◽  
Central Simple ◽  
Suitable Restriction

Let L be an odd degree extension field of the field K, char K ≠ 2. Let U* denote the natural extension map from W(K) to W(L) where W(K), resp. W(L) denotes the Witt group of quadratic forms over K, resp. L. It is well-known that U* is injective [4, p. 198]. In fact Springer [10] proved a stronger theorem, namely that if φ is anisotropic over K then it remains anisotropic on extension to L. Rosenberg and Ware [8] proved that if L is a Galois extension then the image of U* is precisely the subgroup of W(L) fixed by the Galois group of L over K, this Galois group having a natural action on W(L). See [4, p. 214] and [3] for a quick proof. See also Dress [1] who extended these results to equivariant forms. In this article we investigate the corresponding map U* when we replace the field L by a central simple K-algebra of odd degree and indeed more generally by any finite dimensional K-algebra which becomes odd-dimensional on factoring out by the radical. Our algebras are equipped with an involution of the second kind, i.e. one which is non-trivial on the centre, and we replace quadratic forms by hermitian forms with respect to the involution. We show that U* is injective for all the algebras mentioned above and that a weaker version of Springer's theorem holds for central simple algebras of odd degree provided we make a suitable restriction on the nature of the involution. We show that the analogue of the Rosenberg-Ware result is valid for hermitian forms over odd-dimensional Galois field extensions but that for central simple algebras of odd degree a result as nice as the Rosenberg–Ware one cannot hold. Indeed the group of all K-automorphisms of such an algebra which commute with the involution fixes all of the Witt group. However the map U* is not surjective in general even for division algebras of odd degree.

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.

scholarly journals More on the Schur group of a commutative ring

1985 ◽  
Vol 8 (3) ◽  
pp. 513-520
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 earler 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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