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We prove the existence of a set of cardinality 2 n of n -fold Pfister forms over ℂ ( x 1 , … , x n ) which do not share a common ( n − 1 ) -fold factor. This gives a negative answer to a question raised by Becher. The main tools are the existence of the dyadic valuation on the complex numbers and recent results on symmetric bilinear forms over fields of characteristic 2.

Given a monic separable polynomial π of degree 2n over an arbitrary field and a scalar α, we define generic algebras Hπ and $\mathcal {A}_{\alpha \pi }$ for the decomposition of π into a product of two polynomials of degree n and for the factorization απ = 𝜃2 respectively. We investigate representations of degree 1 or 2 of these generic algebras. Every representation of degree 1 of Hπ factors through an etale algebra of degree ${2n}\choose n$ , whereas $\mathcal {A}_{\alpha \pi }$ has no representation of degree 1. We show that every representation of degree 2 of Hπ or $\mathcal {A}_{\alpha \pi }$ factors through the Clifford algebra of some quadratic form, pointed or not, and thus obtain a description of the quaternion algebras that are split by the etale algebra Fπ defined by π of by the function field of the hyperelliptic curve Xαπ with equation y2 = απ(x). We prove that every quaternion algebra split by the function field of Xαπ is also split by Fπ, and provide an example to show that a quaternion algebra split by Fπ may not be split by the function field of any curve Xαπ.

A theorem of Pfister asserts that every $12$-dimensional quadratic form with trivial discriminant and trivial Clifford invariant over a field of characteristic different from $2$ decomposes as a tensor product of a binary quadratic form and a $6$-dimensional quadratic form with trivial discriminant. The main result of the paper extends Pfister's result to orthogonal involutions: every central simple algebra of degree $12$ with orthogonal involution of trivial discriminant and trivial Clifford invariant decomposes into a tensor product of a quaternion algebra and a central simple algebra of degree $6$ with orthogonal involutions. This decomposition is used to establish a criterion for the existence of orthogonal involutions with trivial invariants on algebras of degree $12$, and to calculate the $f_3$-invariant of the involution if the algebra has index $2$.

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.

For a classical group G G of type D n \mathsf {D}_n over a field k k of characteristic different from 2 2 , we show the existence of a finitely generated regular extension of k k over which G G admits outer automorphisms. Using this result and a construction of groups of type A \mathsf {A} from groups of type D \mathsf {D} , we construct new examples of groups of type 2 A n ^2\mathsf {A}_n with n ≡ 3 mod 4 n\equiv 3\bmod 4 and the first examples of type 2 A n ^2\mathsf {A}_n with n ≡ 1 mod 4 n\equiv 1\bmod 4 ( n ≥ 5 ) (n\geq 5) that are not R R -trivial, hence not rational (nor stably rational).

A theorem of Albert-Draxl states that if a tensor product of two quaternion division algebras $Q_1$, $Q_2$ over a field $F$ is not a division algebra, then there exists a separable quadratic extension of $F$ that embeds as a subfield in $Q_1$ and in $Q_2$. We establish a modified version of this result where the tensor product of quaternion algebras is replaced by the corestriction of a single quaternion algebra over a separable field extension. As a tool in the proof, we show that if the transfer of a nonsingular quadratic form $\varphi$ over a quadratic extension is isotropic for a linear functional $s$ such that $s(1)=0$, then $\varphi$ contains a nondegenerate subform defined over the base field.

Homomorphisms are defined between the multiplicative group of an etale algebra of dimension 4 and the multiplicative group of a canonically associated etale algebra of degree 6 over an arbitrary field. These homomorphisms are used to relate various norm groups and to describe the 2-torsion in the relative Brauer group of a separable extension of degree 4. Several remarkable properties of biquadratic extensions are thus extended to arbitrary etale algebras of dimension 4., Comment: in French

Let $\cO$ be a maximal order in a definite quaternion algebra over $\mathbb{Q}$ of prime discriminant $p$, and $\ell$ a small prime. We describe a probabilistic algorithm, which for a given left $O$-ideal, computes a representative in its left ideal class of $\ell$-power norm. In practice the algorithm is efficient, and subject to heuristics on expected distributions of primes, runs in expected polynomial time. This breaks the underlying problem for a quaternion analog of the Charles-Goren-Lauter hash function, and has security implications for the original CGL construction in terms of supersingular elliptic curves., Comment: To appear in the LMS Journal of Computation and Mathematics, as a special issue for ANTS (Algorithmic Number Theory Symposium) conference

An orthogonal involution on a central simple algebra becoming isotropic over any splitting field of the algebra, becomes isotropic over a finite odd degree extension of the base field (provided that the characteristic of the base field is not 2). The proof makes use of a structure theorem for Chow motives with finite coefficients of projective homogeneous varieties, of incompressibility of certain generalized Severi-Brauer varieties, and of Steenrod operations.

According to Tits, the quadric of dimension 6 over the “field” F1 with one element is a set of 8 points structured by a permutation of order 2 without fixed points. Subsets that are disjoint from their image under the permutation are the subspaces of the quadric. As in the classical case of hyperbolic quadrics over an arbitrary field, maximal subspaces come in two different types. We define a geometric triality to be a permutation of order 3 of the set consisting of points and maximal subspaces, carrying points to maximal subspaces of one type and maximal subspaces of the other type to points while preserving the incidence relations. We show analogues over F1 of the one-to-one correspondence between geometric trialities, trialitarian automorphisms of algebraic groups of type D4, and symmetric composition algebras of dimension 8. Here, the algebraic groups of type D4 are replaced by their Weyl group, and composition algebras by a certain type of binary operation on a quadric to which a 0 is adjoined. As in the classical case, we show that there are two types of trialities, one related to octonions and the other to Okubo algebras.

The so-called Tits class, associated to an adjoint absolutely almost simple algebraic group, provides a cohomological obstruction for this group to admit an outer automorphism. If the group has inner type, this obstruction is the only one. In this paper, we prove this is not the case for classical groups of outer type, except for groups of type $^2\mathsf{A}_n$ with $n$ even, or $n=5$. More precisely, we prove a descent theorem for exponent $2$ and degree $6$ algebras with unitary involution, which shows that their automorphism groups have outer automorphisms. In all other relevant classical types, namely $^2\mathsf{A}_n$ with $n$ odd, $n\geq3$ and $^2\mathsf{D}_n$, we provide explicit examples where the Tits class obstruction vanishes, and yet the group does not have outer automorphism. As a crucial tool, we use "generic" sums of algebras with involution.

Author(s): Merkurjev, Alexander S; Tignol, Jean-Pierre | Abstract: For every generalized quadratic form or hermitian form over a division algebra, the anisotropic kernel of the form obtained by scalar extension to the function field of a smooth projective conic is defined over the field of constants. The proof does not require any hypothesis on the characteristic.

Given a central simple algebra with, involution over an arbitrary field, stable subalgebras contained in the space of symmetric elements are investigated. The method emphasizes the similarities between the various types of involutions and privileges a unified treatment for all characteristics whenever possible. As a consequence a conceptual proof of a theorem of Rowen is obtained, which asserts that every division algebra of exponent two and degree eight contains a maximal subfield that is a triquadratic extension of the centre. (C) 2017 Elsevier Inc. A ll rights reserved.

We prove: (1) The group of multipliers of similitudes of a 12-dimensional anisotropic quadratic form over a field K with trivial discriminant and split Clifford invariant is generated by norms from quadratic extensions E/K such that q_E is hyperbolic. (2) If G is the group of K-rational points of an absolutely simple algebraic group whose Tits index is E_{8,2}^{66}, then G is generated by its root groups, as predicted by the Kneser–Tits conjecture.

Severi-Brauer varieties are twisted forms of projective spaces (in the sense of Galois cohomology) and are associated in a functorial way to central simple algebras. Similarly quadrics are related to algebras with involution. Since thin projective spaces are finite sets, thin Severi-Brauer varieties are finite sets endowed with a Galois action; they are associated to etale algebras. Similarly, thin quadrics are etale algebras with involution. We discuss embeddings of thin Severi-Brauer varieties and thin quadrics in Severi-Brauer varieties and quadrics as geometric analogues of embeddings of etale algebras into central simple algebras (with or without involution), and consider the geometric counterpart of the Clifford algebra construction.

We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra Q. We classify these algebras in degree 4 and give an example of such a division algebra with orthogonal involution of degree 8 that does not contain (Q,bar), even though it contains Q and is totally decomposable into a tensor product of quaternion algebras

A quadratic form over a Henselian-valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tame extension. The Witt group of tame quadratic forms is shown to be canonically isomorphic to the Witt group of graded quadratic forms over the graded ring associated to the filtration defined by the valuation, hence also isomorphic to a direct sum of copies of the Witt group of the residue field indexed by the value group modulo 2.

We introduce a type of value function y called a gauge on a finite-dimensional semisimple algebra A over a field F with valuation v. The filtration on A induced by y yields an associated graded ring gr_y(A) which is a graded algebra over the graded field gr_v(F). Key requirements for y to be a gauge are that gr_y(A) be graded semisimple and that dim_(gr_v(F)) (gr_y(A)) = dim_F(A). It is shown that gauges behave well with respect to scalar extensions and tensor products. When v is Henselian and A is central simple over F, it is shown that gr_y(A) is simple and graded Brauer equivalent to gr_w(D), where D is the division algebra Brauer equivalent to A and w is the valuation on D extending v on F. The utility of having a good notion of value function for central simple algebras, not just division algebras, and with good functorial properties, is demonstrated by giving new and greatly simplified proofs of some difficult earlier results on valued division algebras.

Let D be a division algebra of degree 3 over a field containing a primitive cube root of unity. We give two proofs of a theorem of Rost asserting that any two Kummer elements in D can be connected by a chain of length 4.

We define an invariant of torsors under adjoint linear algebraic groups of type C_n-equivalently, central simple algebras of degree 2n with symplectic involution-for n divisible by 4 that takes values in H^3(F, mu_2). The invariant is distinct from the few known examples of cohomological invariants of torsors under adjoint groups. We also prove that the invariant detects whether a central simple algebra of degree 8 with symplectic involution can be decomposed as a tensor product of quaternion algebras with involution.

Elaborating on techniques of Bayer-Fluckiger and Parimala, we prove the following strong version of Serre’s Conjecture II for classical groups: let G be a simply connected absolutely simple group of outer type An or of type Bn, Cn or Dn (non trialitarian) defined over an arbitrary field F. If the separable dimension of F is at most 2 for every torsion prime of G, then every G-torsor is trivial.

An analogue of Springer's theorem on the Witt group of quadratic forms over a complete discretely valued field is proved for Hermitian forms over division algebras over a Henselian field, including some cases where the residue characteristic is 2. Residue forms are defined by means of vector space valuations as Hermitian forms on the graded modules associated with the induced filtrations.

Roughly speaking, the essential dimension of an algebraic structure measures the complexity of the structure by giving the number of independent parameters needed to define it. For a prime number p, the essential p-dimension is a variation of essential dimension that takes into account simplifications in structure that can occur after field extensions of degree prime to p. In §12.1 we define the basic notions of essential dimension and essential p-dimension. We also give examples, which emphasize the relation with the decomposability of central simple algebras into tensor products of symbols. Merkurjev in [150] and Baek–Merkurjev in [21] recognized that the presence of a valuation on a division algebra D implies some level of complexity to D, and they used this to give a lower bound on the essential p-dimension of D. In §12.2 we give a version of the valuation-theoretic part of their arguments. Then in §12.3 we sketch their method to obtain lower bounds on the essential p-dimension of central simple algebras. In the final §12.4, we discuss consequences of these lower bounds for the decomposability of division algebras into tensor products of symbol algebras.

In this chapter we apply the machinery developed in previous chapters to analyze the subfields and splitting fields of division algebras over a Henselian field F. In §9.1 we give properties of the splitting fields of tame division algebra D with center F, with particularly strong criteria proved if D is inertial or totally ramified over F. This leads to explicit constructions of several interesting examples of division algebras, including noncyclic division algebras of degree p 2 with no maximal subfield of the form \(F(\!\sqrt[p^{2}]{a})\) in Examples 9.15, 9.17, and 9.18; noncyclic p-algebras in Ex. 9.26; noncrossed product algebras including universal division algebras in Th. 9.30 and division algebras over Laurent series over \(\mathbb {Q}\), noncrossed products whose degree exceeds the exponent in Cor. 9.46; and crossed product division algebras with only one Galois group for all maximal subfields Galois over the center in Prop. 9.28[9.28].

A central division algebra D over a field F is said to be decomposable if D=D 1⊗ F D 2 for some proper subalgebras D 1, D 2 of D; otherwise, it is indecomposable. In this chapter we give examples of indecomposable algebras, emphasizing constructions that use valuation theory. In light of the primary decomposition, we restrict attention to division algebras of prime power degree. Indecomposable algebras of exponent p 2 or higher are relatively easy to construct as “p-th roots” of other division algebras. Such constructions are discussed in §10.1, where we also give an example of an indecomposable division algebra D that becomes decomposable after a scalar extension of degree prime to \(\operatorname {\mathit{deg}}D\). §10.2 focuses on the more difficult case of indecomposables of prime exponent. We give in §10.2.1 a criterion of Jacob to test the decomposability of a tame semiramified division algebra of prime exponent over a Henselian field. This criterion yields examples of exponent 2 and degree 8 in §10.2.2, and of exponent p≠2 and degree p r with r≥2 in §10.2.4. The last section, §10.3, deals with complete decompositions into tensor products of symbol algebras. The main result is Th. 10.26, which relates armatures in an inertially split division algebra over a Henselian field to armatures in special representatives of its specialization coset.

This chapter is crucial for the study of gauges. Let F be a field with valuation v, and let (F h ,v h ) be the Henselization of (F,v). In §4.1 we prove Morandi’s criterion, Th. 4.1, that v extends to a division algebra D with center F if and only if D⊗ F F h is a division algebra. Among the applications are a primary decomposition theorem and an Ostrowski-type defect theorem, Th. 4.3, for valued division algebras. The valuation v is said to be defectless in D if equality holds in the Fundamental Inequality for the extension of v to D. In §4.2 we extend the notion of defectlessness of v to any semisimple F-algebra A by considering the simple components of A⊗ F F h . In §4.3 we prove key classification results for gauges. We show in Th. 4.26 that if S is a central simple F h -algebra, then any v h -gauge on S is an \(\operatorname {\mathit{End}}\)-gauge as in Prop. 3.34. We also show how v-gauges on a semisimple F-algebra A are built from gauges on the simple components A i of A for the extensions of v to Z(A i ). §4.4 is devoted to proving Th. 4.50, which says that a semisimple F-algebra has a v-gauge if and only if v is defectless in A.

We pursue in this chapter the investigation of valuations through the graded structures associated to the valuation filtration. The graded field of a valued field is an enhanced version of the residue field, inasmuch as it encapsulates information about the value group in addition to the residue field. It thus captures much of the structure of the field, particularly in the Henselian case. This point is made clear in §5.2, where we show that—when the ramification is tame—Galois groups and their inertia subgroups of Galois extensions of valued fields can be determined from the corresponding extension of graded fields. Henselian fields are shown to satisfy a tame lifting property from graded field extensions, generalizing the inertial lifting property. In §5.1, we lay the groundwork for the subsequent developments by an independent study of graded fields, their algebraic extensions and their Galois theory.

In this chapter we introduce the central object of study in this book: valuations on a division algebra D finite-dimensional over its center F. In §1.1 we define valuations and describe the associated structures familiar from commutative valuation theory: the valuation ring \(\mathcal {O}_{D}\), its unique maximal left and maximal right ideal \(\mathfrak {m}_{D}\), the residue division algebra \(\overline{D}\), and the value group Γ D . We also describe an important and distinctively noncommutative feature, namely a canonical homomorphism θ D from Γ D to the automorphism group \(\operatorname {\mathit{Aut}}(Z(\overline{D})\big/\,\overline{F}\,)\); θ D is induced by conjugation by elements of D ×. In §1.2, after proving the “Fundamental Inequality for valued division algebras, we look at valuations on D from the perspective of F. We show that a valuation on F has at most one extension to D, and prove a criterion for when such an extension exists. When this occurs, we show that the field \(Z(\overline{D})\) is finite-dimensional and normal over \(\overline{F}\) and that θ D is surjective. We also describe the technical adjustments needed to apply the classical method of “composition” of valuations to division algebras. The filtration on D induced by a valuation leads to an associated graded ring \(\operatorname {\mathsf {gr}}(D)\), which we describe in §1.3. Throughout the book we emphasize use of \(\operatorname {\mathsf {gr}}(D)\) to help understand the valuation on D. This chapter includes many examples of division algebras with valuations.

The trace form of a central simple algebra of degree 4. Let k be a field of characteristic different from 2 containing a primitive 4-th root of unity. We show that the trace quadratic form of any central simple k-algebra A of degree 4 decomposes in the Witt group of k as the sum of a 2-fold Pfister form q(2) and a 4-fold Pfister form q(4) which are uniquely determined by A. The form q2 is the norm form of the quaternion algebra Brauer-equivalent to A circle times(k) A, and q(4) is hyperbolic if and only if A is a symbol algebra. To cite this article: M. Rost et al., C R. Acad. Sci. Paris, Ser. 1342 (2006). (c) 2005 Academie des sciences. Publie par Elsevier SAS. Tons droits reserves.

Let A be a central simple algebra of degree 4 over a field k of characteristic 2 and let q(A) be the quadratic form on A given by the second coefficient of the reduced characteristic polynomial. We show that A uniquely determines a 2-fold Pfister form q_2 and a 4-fold Pfister form q_4 such that q(A) = [1, 1] + q_2 + q_4 in the Witt group of k, where [1, 1] is the form x^2 + xy + y^2. The form q_2 is the norm form of the quaternion algebra Brauer-equivalent to A^2, and q_4 is hyperbolic if and only if A is cyclic.

In this note we give examples of division rings which posses an anti-automorphism but no involution. The motivation for such examples comes from geometry. If D is a division ring and V a finite-dimensional right D-vector space of dimension ≥ 3, then the projective geometry P(V ) has a duality (resp. polarity) if and only if D has an anti-automorphism (resp. involution) [2, p. 97, p. 111]. Thus, the existence of a division ring with an anti-automorphism but no involution gives examples of projective geometries with dualities but no polarities. All the division rings considered in this paper are finite-dimensional algebras over their center. The anti-automorphisms we construct are not linear over the center (i.e. they do not restrict to the identity on the center) since a theorem of Albert [1, Theorem 10.19] shows that every finite-dimensional central division algebra with a linear anti-automorphism has an involution. Several proofs of this result can be found in the literature, see for instance [5] or [10, Chapter 8, §8]. The paper is organized as follows. Section 2 collects some background information on central division algebras. Section 3 establishes the existence of division algebras over algebraic number fields which have anti-automorphisms but no involution, and gives two explicit constructions of such algebras. The proofs rely on deep classical results on the Brauer group of number fields. Section 4 gives examples whose center has a more complicated structure (they are Laurent series fields over local fields), but the proof that these algebras have no involution is more elementary. Sections 3 and 4 are independent of each other.

An invariant for symplectic involutions on central simple algebras of degree divisible by 4 over fields of characteristic different from 2 is defined on the basis of Rost's cohomological invariant of degree 3 for torsors under symplectic groups. We relate this invariant to trace forms and show how its triviality yields a decomposability criterion for algebras of degree 8 with symplectic involution.

Let D D be a division algebra of prime degree p p . A set of criteria is given for cyclicity of D D in terms of subgroups of the multiplicative group D ∗ D^* of D D . It is essentially shown that D D is cyclic if and only if D ∗ D^* contains a nonabelian metabelian subgroup.

The centers of the generic central simple algebras with involution are interesting objects in the theory of central simple algebras. These fields also arise as invariant fields for linear actions of projective orthogonal or symplectic groups. In this paper, we prove that when the characteristic is not 2, these fields are retract rational, in the case the degree is $8m$ and $m$ is odd. We achieve this by proving the equivalent lifting property for the class of central simple algebras of degree $8m$ with involution. A companion paper ([S3]) deals with the case of $m$, $2m$ and $4m$ where stronger rationality results are proven., Comment: 7 pages

The centralizer of a square-central skew-symmetric unit in a central simple algebra with orthogonal involution carries a unitary involution. The discriminant algebra of this unitary involution is shown to be an orthogonal summand in one of the components of the Clifford algebra of the orthogonal involution. As an application, structure theorems for orthogonal involutions on central simple algebras of degree 8 are obtained.

Quadratic forms over division algebras over local or global fields of characteristic 2 are classified by an invariant derived from the Clifford algebra construction.