Generic Splitting Fields Research Articles

The $J$-invariant over splitting fields of Tits algebras

We describe the $J$-invariant of a semisimple algebraic group $G$ over a generic splitting field of a Tits algebra of $G$ in terms of the $J$-invariant over the base field. As a consequence we prove a 10-year-old conjecture of Quéguiner-Mathieu, Semenov, and Zainoulline on the $J$-invariant of groups of type $\mathrm {D}_n$. In the case of type $\mathrm {D}_n$ we also provide explicit formulas for the first component and in some cases for the second component of the $J$-invariant.

A remark on a theorem of Bavula on Lüroth extensions

Using a theorem of Roquette–Ohm [P. Roquette, Isomorphisms of generic splitting fields of simple algebras, J. Reine Angew. Math. 214–215 (1964) 207–226 and J. Ohm, On subfields of rational function fields, Arch. Math. 42 (1984) 136–138], we indicate a short proof that a rational extension of a Lüroth extension of a field k is a Lüroth extension of k . This assertion was proved by Bavula [V. Bavula, Lüroth field extensions, J. Pure Appl. Algebra 199 (2005) 1–10] with the added hypothesis that char ( k ) = 0 .

Essential dimension of quadrics

Let X be an anisotropic projective quadric over a field F of characteristic not 2. The essential dimension dimes(X ) of X ,a s def ined by Oleg Izhboldin, is dimes(X ) = dim(X ) − i(X ) + 1 , where i(X ) is the first Witt index of X (i.e., the Witt index of X over its function field). Let Y be a complete (possibly singular) algebraic variety over F with all closed points of even degree and such that Y has a closed point of odd degree over F(X ). Our main theorem states that dimes(X ) ≤ dim(Y ) and that in the case dimes(X ) = dim(Y ) the quadric X is isotropic over F(Y ). Applying the main theorem to a projective quadric Y , we get a proof of Izhboldin's conjecture stated as follows: if an anisotropic quadric Y becomes isotropic over F(X ) ,t hen dim es(X ) ≤ dimes(Y ), and the equality holds if and only if X is isotropic over F(Y ). We also solve Knebusch's problem by proving that the smallest transcendence degree of a generic splitting field of a quadric X is equal to dimes(X ). Let (V ,ϕ ) be a non-degenerate quadratic form of dimension at least 2 over a field F of characteristic not 2 and let X = Q(ϕ) be the quadric hypersurface given by the equation ϕ(x) = 0 in the projective space P(V ). We say that the quadric X is anisotropic if ϕ is an anisotropic quadratic form. By Springer's theorem, every closed point of an anisotropic quadric X has

Formes antihermitiennes devenant hyperboliques sur un corps de déploiement

Let H=( a, b) F be a division quaternion algebra over a field F of characteristic not 2. Denote by τ the canonical involution on H and by K a splitting field of H. If h is a skew-hermitian form over ( H, τ) then, by extension of scalars to K and by Morita equivalence, we obtain a quadratic form h K over K. This gives a map of Witt groups ρ:W −1( H, τ)→W( K) induced by ρ( h)= h K . When K is a generic splitting field of H we prove in this note that the map ρ is injective.

Generic Splitting Fields of Central Simple Algebras: Galois Cohomology and Nonexcellence

A field extension L / F is called excellent if, for any quadratic form φ over F, the anisotropic part (φL)an of φ over L is defined over F; L / F is called universally excellent if L ⋅ E / E is excellent for any field extension E / F. We study the excellence property for a generic splitting field of a central simple F-algebra. In particular, we show that it is universally excellent if and only if the Schur index of the algebra is not divisible by 4. We begin by studying the torsion in the second Chow group of products of Severi–Brauer varieties and its relationship with the relative Galois cohomology group H3(L / F) for a generic (common) splitting field L of the corresponding central simple F-algebras.

The index of a Brauer class on a Brauer-Severi variety

Let D D and E E be central division algebras over k k ; let K K be the generic splitting field of E E ; we show that the index of D ⊗ k K D{ \otimes _k}K is the minimum of the indices of D ⊗ E ⊗ i D \otimes {E^{ \otimes i}} as i i varies. We use this to calculate the index of D D under related central extensions and to construct division algebras with special properties.

Constructions of Brauer-Severi Varieties and Norm Hypersurfaces

Let k be any field, A a central simple k-algebra of degree m (i.e., dimk A = m2). Several methods of constructing the generic splitting fields for A are proposed and Saltman proves that these methods result in almost the same generic splitting field [8, Theorems 4.2 and 4.4]. In fact, the generic splitting field constructed by Roquette [7] is the function field of the Brauer- Severi variety Vm(A) while the generic splitting field constructed by Heuser and Saltman [4 and 8] is the function field of the norm surface W(A). In this paper, to avoid possible confusion about the dimension, we shall call it the norm hypersurface instead of the norm surface.

The centers of generic division algebras with involution

The main goal of this paper is a study of the centers of the generic central simple algebras with involution. These centers are shown to be invariant fields under finite groups in a way analagous to the center of the generic division algebras. The centers of the generic central simple algebras with involution are also described as generic splitting fields (i.e. function fields of Brauer-Severi varieties) over the centers of generic division algebras. Finally, a generic central simple algebra is described for the class of central simple algebras with subfields of a certain dimension.

Generic reducing fields of Jordan pairs

Generic reducing fields of Jordan pairs, generalizing at the same time generic splitting fields of associative algebras and generic zero fields of quadratic forms, are intrinsically defined and constructed. The most elementary properties are derived, and the relationship with other generic constructions, particularly those linked to Brauer-Severi varieties, are investigated. As an application it is shown that there exist nonisomorphic exceptional Jordan division algebras having the same splitting fields.

Generic splitting fields

(1978). Generic splitting fields. Communications in Algebra: Vol. 6, No. 10, pp. 1017-1035.

Generic splitting fields of composition algebras

Witt [7] proved that one can assign to each generalized quaternion algebra -' over a field K, a field F(sQI) containing K which splits S( and has the property: if F(a) splits a quaternion algebra 4 over K then either M is split over K or V is isomorphic to a. Amitsur [2] has generalized this result to obtain generic splitting fields for all central simple associative algebras of dimension greater than one over K (cf. Roquette [6]). In this paper we generalize the result of Witt in another direction, studying splitting fields of composition algebras of dimension greater than one over K of characteristic other than two. We assign to each such algebra ', a field F(e) containing K, prove that F(W) is an invariant under isomorphisms, and prove

On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras

On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras