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We introduce a presentation for central simple algebras over a field k using Amitsur cohomology. We provide efficient algorithms for computing a cocycle corresponding to any such algebra given by structure constants. If k is a number field, we use this presentation to prove that the explicit isomorphism problem (i.e., finding an isomorphism between central simple algebras given by structure constants) reduces to S -unit group computation and other related number theoretical computational problems. This also yields, conditionally on the generalised Riemann hypothesis, the first polynomial quantum algorithm for the explicit isomorphism problem over number fields. [ABSTRACT FROM AUTHOR]

An associative central simple algebra is a form of a matrix algebra, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrix algebras, which we are led to define and investigate. Semiassociative algebras modulo skew matrix algebras compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup. [ABSTRACT FROM AUTHOR]

We propose a new identification system based on algorithmic problems related to computing isomorphisms between central simple algebras. We design a statistical zero knowledge protocol which relies on the hardness of computing isomorphisms between orders in division algebras which generalizes a protocol by Hartung and Schnorr, which relies on the hardness of integral equivalence of quadratic forms. [ABSTRACT FROM AUTHOR]

The goal of this note is to reduce the existing upper bound for the essential dimension of central simple algebras of degree 8 and exponent 2 over fields of characteristic 2 from 10 to 9. [ABSTRACT FROM AUTHOR]

We extend, to the case of local fields of even characteristic, our previous computations for the set of maximal orders containing given quaternions as concrete sub-graphs of the Bruhat–Tits tree. Computing the relative position of such sub-graphs, the branches, is useful, for instance, to explicitly compute relative spinor images, thus solving the selectivity problem. They also play a role in the description of quotient graphs, which are useful to study matrix groups of arithmetical importance. As in earlier work, we concentrate on orders generated by two quaternions, but extending our method to larger generating sets is quite straightforward. In our previous work, the results were given mainly in terms of the quadratic defect. In the present context, we introduce and characterize an analogous concept for Artin–Scheier extensions, to take care of quaternions generating Galois field extensions. The latter extensions have no non-trivial elements of null trace. For this reason, we state our results in terms of a wider family of algebra presentations, in contrast to the aforementioned work, where we chose a pair of pure quaternions as generators of the order. [ABSTRACT FROM AUTHOR]

Let RG be the group ring of a finite group G over a commutative ring R with 1. An element x in RG is said to be skew-symmetric with respect to an involution σ of RG if σ (x) = - x. A structure theorem for the Lie algebra of skew-symmetric elements of FG is given where F is an algebraic extension of Q which generalizes some previously known results in this direction. [ABSTRACT FROM AUTHOR]

We show that any adjoint absolutely simple linear algebraic group over a field of characteristic zero is the automorphism group of some projector on a central simple algebra. Projective homogeneous varieties can be described in these terms; in particular, we reproduce quadratic equations by Lichtenstein defining them. [ABSTRACT FROM AUTHOR]

Given a field F of characteristic 3 and division symbol p-algebras [ α , β) 3 , F and [ α , γ) 3 , F of degree 3 over F, we prove that if α dlog (β) ∧ dlog (γ) is trivial in the Kato–Milne cohomology group H 3 3 (F) , then the algebras share a common splitting field which is an inseparable degree 3 extension of either F or a quadratic extension of F. In the special case of quadratically closed fields, if α dlog (β) ∧ dlog (γ) = 0 , then they share an inseparable degree 3 extension of F. [ABSTRACT FROM AUTHOR]

We describe the set of maximal orders in a 2-by-2 matrix algebra over a non-commutative local division algebra B containing a given suborder, for certain important families of such suborders, including rings of integers of division subalgebras of B or most maximal semisimple commutative subalgebras. [ABSTRACT FROM AUTHOR]

Abstract We compute the group of K 1 -zero-cycles on the second generalized involution variety for an algebra of degree 4 with symplectic involution. This description is given in terms of the group of multipliers of similitudes associated to the algebra with involution. Our method utilizes the framework of Chernousov and Merkurjev for computing K 1 -zero-cycles in terms of R -equivalence classes of prescribed algebraic groups. This gives a computation of K 1 -zero-cycles for some homogeneous varieties of type C 2. [ABSTRACT FROM AUTHOR]

Algebraic space-time coding—a powerful technique developed in the context of multiple-input multiple-output (MIMO) wireless communications—has profited tremendously from tools from Class Field Theory and, more concretely, the theory of central simple algebras and their orders. During the last decade, the study of space-time codes for practical applications, and more recently for future generation (5G+) wireless systems, has provided a practical motivation for the consideration of many interesting mathematical problems. One such problem is the explicit computation of orders of central simple algebras with small discriminants. In this article, we consider the most interesting asymmetric MIMO channel setups and, for each treated case, we provide explicit pairs of fields and a corresponding non-norm element giving rise to a cyclic division algebra whose natural order has the minimum possible discriminant. [ABSTRACT FROM AUTHOR]

An orthogonal involution σ on a central simple algebra A , after scalar extension to the function field F ( A ) of the Severi–Brauer variety of A , is adjoint to a quadratic form q σ over F ( A ) , which is uniquely defined up to a scalar factor. Some properties of the involution, such as hyperbolicity, and isotropy up to an odd-degree extension of the base field, are encoded in this quadratic form, meaning that they hold for the involution σ if and only if they hold for q σ . As opposed to this, we prove that there exists non-totally decomposable orthogonal involutions that become totally decomposable over F ( A ) , so that the associated form q σ is a Pfister form. We also provide examples of nonisomorphic involutions on an index 2 algebra that yield similar quadratic forms, thus proving that the form q σ does not determine the isomorphism class of σ , even when the underlying algebra has index 2. As a consequence, we show that the e 3 invariant for orthogonal involutions is not classifying in degree 12, and does not detect totally decomposable involutions in degree 16, as opposed to what happens for quadratic forms. [ABSTRACT FROM AUTHOR]

The u -invariant of a field is the supremum of the dimensions of anisotropic quadratic forms over the field. We define corresponding u -invariants for hermitian and generalised quadratic forms over a division algebra with involution in characteristic 2 and investigate the relationships between them. We investigate these invariants in particular in the case of a quaternion algebra and further when this quaternion algebra is the unique quaternion division algebra over a field. [ABSTRACT FROM AUTHOR]

In this paper we show that there exist rational maps between varieties associated to isomorphism classes of central simple algebras such as Severi–Brauer varieties, norm hypersurfaces and symmetric powers, provided the Brauer-equivalence classes of the algebras involved generate the same cyclic subgroup in Br ( k ) . In some cases we even get rational embeddings. We relate the obtained results to the Amitsur conjecture. [ABSTRACT FROM AUTHOR]

We discuss the Kummer subspaces of tensor products of cyclic algebras, focusing mainly on the case of cyclic algebras of degree 3. We present a family of maximal Kummer spaces in any tensor product of cyclic algebras of prime degree, classify all the monomial Kummer spaces in tensor products of cyclic algebras of degree 3, and provide an upper bound for the dimension of Kummer spaces in the generic tensor product of cyclic algebras of degree 3. [ABSTRACT FROM AUTHOR]

We prove that if the maximal dimension of an anisotropic homogeneous polynomial form of prime degree over a field with is a finite integer greater than 1 then the symbol length of -algebras of exponent over is bounded from above by , and show that every two tensor products of symbol algebras of lengths and with can be modified so that they share a common slot. For , we obtain an upper bound of for the symbol length, which is sharp when . [ABSTRACT FROM AUTHOR]

In this manuscript, it is shown that the group of K 1 -zero-cycles on the second generalized Severi–Brauer variety of an algebra A of index 4 is given by elements of the group K 1 ( A ) together with a square-root of their reduced norm. Utilizing results of Krashen concerning exceptional isomorphisms, we translate our problem to the computation of cycles on involution varieties. Work of Chernousov and Merkurjev then gives a means of describing such cycles in terms of Clifford and spin groups and corresponding R -equivalence classes. We complete our computation by giving an explicit description of these algebraic groups. [ABSTRACT FROM AUTHOR]

A representation field for a non-maximal order H in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders representing H. In our previous work we have proved the existence of the representation field for several important families of suborders, like commutative orders, while we have also found examples where the representation field fails to exist. To be precise, we have found full-rank orders, in central simple algebras of dimension 9 or larger over a suitable field, whose representation field is undefined. In this article, we prove that the representation field is defined for any order H of rank r ≤ 7. This is done by defining representation fields for arbitrary representations of orders into central simple algebra and showing that the computation of these generalized representation fields can be reduced to the case of irreducible representations. The same technique yields the existence of representation fields for any order in an algebra whose semi-simple reduction is commutative. We also construct a rank-8 order, in a 16-dimensional matrix algebra, whose representation field is not defined. [ABSTRACT FROM AUTHOR]

We prove that the maximal dimension of a p-central subspace of the generic symbol p-algebra of prime degree p is $${p+1}$$ . We do it by proving the following number theoretic fact: let $${\{s_1,\dots,s_{p+1} \}}$$ be $${p+1}$$ distinct nonzero elements in the additive group $${G=(\mathbb{Z}/p \mathbb{Z}) \times (\mathbb{Z}/p \mathbb{Z})}$$ , then every nonzero element $${g \in G}$$ can be expressed as $${d_1 s_1+\dots+d_{p+1} s_{p+1}}$$ for some non-negative integers $${d_1,\dots,d_{p+1}}$$ with $${d_1+\dots+d_{p+1} \leq p-1}$$ . [ABSTRACT FROM AUTHOR]

Let G be a split simple affine algebraic group of type A or C over a field k , and let E be a standard generic G -torsor over a field extension of k . We compute the Chow ring of the variety of Borel subgroups of G (also called the variety of complete flags of G ), twisted by E . In most cases, the answer contains a large finite torsion subgroup. The torsion-free cases have been treated in the predecessor Chow ring of some generically twisted flag varieties by the author. [ABSTRACT FROM AUTHOR]

We compute the motive of the algebraic group G = SL1(D) for a central simple algebra D of prime degree over a perfect field. As an application we determine certain motivic cohomology groups and differentials in the motivic spectral sequence of G. [ABSTRACT FROM AUTHOR]

In this paper we show that a split central simple algebra with quadratic pair which decomposes into a tensor product of quaternion algebras with involution and a quaternion algebra with quadratic pair is adjoint to a quadratic Pfister form. This result is new in characteristic two, otherwise it is equivalent to the Pfister Factor Conjecture proven in Becher (Invent Math 173:1-6, 2008). [ABSTRACT FROM AUTHOR]

In this article, the first general constructions of fast-decodable, more specifically (conditionally) $g$-group decodable, space–time block codes for the nonorthogonal amplify and forward (NAF) multiple-input multiple-output (MIMO) relay channel under the half-duplex constraint are proposed. In this scenario, the source and the intermediate relays used for data amplification are allowed to employ multiple antennas for data transmission and reception. The worst-case decoding complexity of the obtained codes is reduced by up to 75%. In addition to being fast-decodable, the proposed codes achieve full-diversity and have nonvanishing determinants, which has been shown to be useful for achieving the optimal diversity-multiplexing tradeoff (DMT) of the NAF channel. Furthermore, it is shown that the same techniques as in the cooperative scenario can be utilized to achieve fast-decodability for $K$-user MIMO multiple-access channel (MAC) space–time block codes. The resulting codes in addition exhibit the conditional nonvanishing determinant property which, for its part, has been shown to be useful for achieving the optimal MAC-DMT. [ABSTRACT FROM PUBLISHER]

We show that a non-hyperbolic quadratic pair on a central simple algebra Brauer equivalent to a quaternion algebra stays non-hyperbolic over some splitting field of the quaternion algebra. This extends a result previously only known for fields of characteristic different from two. Our presentation is free from restrictions on the characteristic of the base field. [ABSTRACT FROM AUTHOR]

Résumé Soit K le corps des fonctions d'une courbe projective lisse X sur un corps local supérieur k . On exhibe des exemples et des contre-exemples au principe local–global pour les algèbres simples centrales sur K , qui est une hypothèse importante pour disposer de suites exactes de Poitou–Tate pour les tores sur K . Let K be the function field of a smooth projective curve X over a higher-dimensional local field k . We give examples and counter-examples to the local–global principle for central simple algebras over K , which is an important hypothesis for having Poitou–Tate exact sequences for tori over K . [ABSTRACT FROM AUTHOR]

In this article, we present a decomposition theorem for unitary spaces over a central simple division algebra with involution in characteristic 2. This is a generalization of a decomposition result for quadratic forms in characteristic 2 from [4] and extends a generalization of the Witt decomposition theorem for nonsingular spaces in this setting to cover spaces that may be singular. [ABSTRACT FROM AUTHOR]

We prove that if two division p-algebras of prime degree share an inseparable field extension of the center then they also share a cyclic separable one. We show that the converse is in general not true. We also point out that sharing all the inseparable field extensions of the center does not imply sharing all the cyclic separable ones. [ABSTRACT FROM AUTHOR]

Let k be a field, K/k he a quadratic separable field extension, and A a finite dimensional central simple algebra over K. If k is global or the field of fractions of a two-dimensional excellent henselian local domain with an algebraically dosed residue field of characteristic zero und the degree of A is odd, we prove that all K/k-involutions on A are cyclic. [ABSTRACT FROM AUTHOR]

In this note we prove a result comparing rationality of algebraic cycles over the function field of a SL 1 ( A ) -torsor for a central simple algebra A and over the base field. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic 2, and T ( F ) the set of totally singular F -quadratic forms up to isometry. The u ˆ -invariant of F is the maximal dimension of an anisotropic F -quadratic form. In this paper we introduce new invariants of F related to the u ˆ -invariant. The first one, called u ˜ -invariant, is the maximal dimension of an anisotropic not totally singular F -quadratic form. The motivation of introducing this invariant is to see how the values of the u ˆ -invariant can be given without the use of forms of T ( F ) , this is because most of the known values of the u ˆ -invariant are realized by the forms of T ( F ) (see [12] ). Also, for r , s ≥ 1 integers, we introduce the u r -invariant ( resp. the u ˜ s -invariant) which is the maximal dimension of an anisotropic F -quadratic form having a regular part of dimension 2 r ( resp. the maximal dimension of an anisotropic not totally singular F -quadratic form having a quasilinear part of dimension s ). We make a comparison between these new invariants by relating them to the u -invariant and the u ˆ -invariant, and we give a list of values that they can take. We also discuss the classical question, due to Baeza [3] , on the universality of some nonsingular F -quadratic forms when F has finite u -invariant. [ABSTRACT FROM AUTHOR]

We generalize the result of Karpenko and Reichstein () on incompressibility of Galois Weil transfer of generalized Severi-Brauer varieties to direct products of varieties of such type; as shown in Karpenko and Reichstein (), this is needed to compute essential dimension of representations of finite groups. We also provide a generalization to non-Galois (separable) Weil transfer. [ABSTRACT FROM AUTHOR]

We extend the method of Cassels for computing the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank of an elliptic curve obtained by 3-descent. [ABSTRACT FROM AUTHOR]

We define a notion of Morita equivalence between algebras with antiautomorphisms such that two equivalent algebras have the same category of sesquilinear forms. This generalizes the Morita equivalence of algebras with involutions defined by Fröhlich and Mc Evett [5], and their categories of ϵ-hermitian forms. For two Morita equivalent algebras with involution, with an additional technical property (which is true for central simple algebras), we define a new algebra with antiautomorphism, called the orthogonal sum, which generalizes the usual notion of orthogonal sum of forms. We explore the invariants of this sum. [ABSTRACT FROM PUBLISHER]

The main result of this paper states that if $k$ is a field of characteristic $p\gt 0$ and $A/ k$ is a central simple algebra of index $d= {p}^{n} $ and exponent ${p}^{e} $, then $A$ is split by a purely inseparable extension of $k$ of the form $k( \sqrt[{p}^{e} ]{{a}_{i} }, i= 1, \ldots , d- 1)$. Combining this result with a theorem of Albert (for which we include a new proof), we get that any such algebra is Brauer equivalent to the tensor product of at most $d- 1$ cyclic algebras of degree ${p}^{e} $. This gives a drastic improvement upon previously known upper bounds. [ABSTRACT FROM AUTHOR]

In this paper we characterize involutions that become metabolic over a quadratic field extension attained by adjoining a square root. [ABSTRACT FROM AUTHOR]

Abstract: Let D be a central division algebra over a field F. We study in this note the rigidity of the motivic decompositions of the Severi–Brauer varieties of D, with respect to the ring of coefficients and to the base field. We first show that if the ring of coefficient is a field, these decompositions only depend on its characteristic. In a second part we show that if D remains division over a field extension , the motivic decompositions of several Severi–Brauer varieties of D remain the same when extending the scalars to E. [Copyright &y& Elsevier]

Abstract: If are two orders in a central simple algebra with of maximal rank, the theory of representation fields describes the set of spinor genera of orders in the genus of representing the order . When is contained in a maximal subfield of and the dimension of is the square of a prime p, the proportion of spinor genera representing has the form . In fact, when the representation field exists, this proportion is either 1 or . In the later case the order is said to be selective for the genus. The condition for selectivity is known when is maximal and also when and is an Eichler order. In this work we describe the orders that are selective for at least one genus of orders of maximal rank in . [Copyright &y& Elsevier]

Abstract: For an n-dimensional central simple algebra defined by a generalized Hilbert symbol over a number field, we compute the number of conjugacy classes of maximal orders that contain a conjugate of the order spanned by the canonical generators of the algebra. This has applications to the study of certain finite subgroups of arithmetic orbifold groups. [Copyright &y& Elsevier]

In this paper, we prove a result of Rost, which describes the cohomological invariants of central simple algebras of degree 4 with values in μ when the base field contains a square root of −1. [ABSTRACT FROM AUTHOR]

Abstract: 1. Let be central simple disjoint algebras over a field F. Let also , , , and for each , let and have the same sets of prime divisors. Then there exists a field extension such that and , . 2. Let be a central simple algebra over a field K with an involution τ of the second kind. We prove that there exists a regular field extension preserving indices of central simple K-algebras such that is cyclic and has an involution of the second kind extending τ. [Copyright &y& Elsevier]

We give some conditions under which no two non-conjugate projective representations, in an algebra, of a given group can become conjugate over a separable extension of the base field. In particular, we show this is always the case for groups with trivial abelianization. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the conditions under which an involution becomes metabolic over a quadratic field extension. We characterise those involutions that become metabolic over a given separable quadratic extension. We further give an example of an anisotropic orthogonal involution that becomes isotropic over a separable quadratic extension. [Copyright &y& Elsevier]

Abstract: Fixing a field F of characteristic different from 2 and 3, we consider pairs consisting of a degree 3 central simple F-algebra A and a 3-dimensional vector subspace V of the reduced trace zero elements of A which is totally isotropic for the trace quadratic form. Each such pair gives rise to a cubic form mapping an element of V to its cube; therefore we call it a cubic pair over F. Using the Okubo product in the case where F contains a primitive cube root of unity, and extending scalars otherwise, we give an explicit description of all isomorphism classes of such pairs over F. We deduce that a cubic form associated with an algebra in this manner determines the algebra up to (anti-)isomorphism. [Copyright &y& Elsevier]

Perfect space—time codes are codes for the coherent multiple-input multiple-output (MIMO) channel. They have been called so since they satisfy a large number of design criteria that makes their performances outmatch many other codes. In this correspondence, we discuss the existence of such codes (or more precisely, the existence of perfect codes with optimal signal complexity). [ABSTRACT FROM AUTHOR]

Abstract: The Bröcker-Prestel local-global principle characterizes weak isotropy of quadratic forms over a formally real field in terms of weak isotropy over the Henselizations and isotropy over the real closures of that field. A Hermitian analogue of this principle is presented for algebras of index at most two. An improved result is also presented for algebras with a decomposable involution, algebras of Pythagorean index at most two, and algebras over SAP and ED fields. [Copyright &y& Elsevier]

We give a cohomological characterization of the set of conjugacy classes of finite subgroups of the projective multiplicative group of a finite dimensional algebra that become conjugate to a given group over some finite separable extension of the base field. We provide two applications. [ABSTRACT FROM AUTHOR]