Separable Quadratic Extension Research Articles

The Pentagon Theorem in Miquelian Möbius Planes

We give an algebraic proof of the Pentagon Theorem. The proof works in all Miquelian Möbius planes obtained from a separable quadratic field extension. In particular, the theorem holds in every finite Miquelian plane. The arguments also reveal that the five concyclic points in the Pentagon Theorem are either pairwise distinct or identical to one single point. In addition we identify five additional quintuples of points in the pentagon configuration which are concyclic.

The Hidden Twin of Morley’s Five Circles Theorem

We give an algebraic proof of a slightly extended version of Morley’s Five Circles Theorem. The theorem holds in all Miquelian Möbius planes obtained from a separable quadratic field extension, in particular in the classical real Möbius plane. Moreover, the calculations bring to light a hidden twin of the Five Circles Theorem that seems to have been overlooked until now.

Explicit isomorphisms of quaternion algebras over quadratic global fields

Let L be a separable quadratic extension of either {mathbb {Q}} or {mathbb {F}}_q(t). We exhibit efficient algorithms for finding isomorphisms between quaternion algebras over L. Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple L-algebra.

Discrete series and the essentially Speh representations

Let π denote an essentially Speh representation of the general linear group over a non-archimedean local field or its separable quadratic extension, and let σc denote an irreducible cuspidal representation of either symplectic, special odd-orthogonal, or unitary group. We determine when the induced representation π⋊σc contains a discrete series subquotient. We also identify all discrete series subquotients.

Linear spaces embedded into projective spaces via Baer sublines

Every nontrivial linear space embedded in a Pappian projective space such that the blocks of the linear space are projectively equivalent Baer sublines with respect to a separable quadratic field extension is either a Baer subspace, or a Hermitian unital.Mathematics Subject Classifications: 51A45, 51E10

The Clemens–Griffiths method over non-closed fields

We use the Clemens-Griffiths method to construct smooth projective threefolds, over any field $k$ admitting a separable quadratic extension, that are $k$-unirational and $\bar{k}$-rational but not $k$-rational. When $k=\mathbb{R}$, we can moreover ensure that their real locus is diffeomorphic to the real locus of a smooth projective $\mathbb{R}$-rational variety and that all their unramified cohomology groups are trivial.

Cohomological kernels of mixed extensions in characteristic 2

Let F be a field of characteristic 2, ΩFm the space of m-differential forms over F, d:ΩFm−1⟶ΩFm the differential operator, and H2m+1(F) the cokernel of the Artin-Schreier operator ℘:ΩFm⟶ΩFm/dΩFm−1 given on generators by: ℘(xdx1x1∧⋯∧dxmxm)=(x2−x)dx1x1∧⋯∧dxmxm+dΩFm−1. It was shown in [8] that given a multiquadratic purely inseparable extension L/F and a separable quadratic extension K/F, then the kernel of the natural homomorphism H2m+1(F)⟶H2m+1(K⋅L) is equal to the sum of the kernels of the extensions L/F and K/F. In this paper we extend this result to the compositum of a finite purely inseparable multiquadratic extension with a separable biquadratic extension, and we prove that this splitting property of kernels is no longer true by computing the kernel of the compositum of a separable quadratic (or biquadratic) extension with a simple purely inseparable extension of arbitrary degree.

Average values of L-functions in even characteristic

Let k=Fq(T) be the rational function field over a finite field Fq, where q is a power of 2. In this paper we solve the problem of averaging the quadratic L-functions L(s,χu) over fundamental discriminants. Any separable quadratic extension K of k is of the form K=k(xu), where xu is a zero of X2+X+u=0 for some u∈k. We characterize the family I (resp. F, F′) of rational functions u∈k such that any separable quadratic extension K of k in which the infinite prime ∞=(1/T) of k ramifies (resp. splits, is inert) can be written as K=k(xu) with a unique u∈I (resp. u∈F, u∈F′). For almost all s∈C with Re(s)≥12, we obtain the asymptotic formulas for the summation of L(s,χu) over all k(xu) with u∈I, all k(xu) with u∈F or all k(xu) with u∈F′ of given genus. As applications, we obtain the asymptotic mean value formulas of L-functions at s=12 and s=1 and the asymptotic mean value formulas of the class number hu or the class number times regulator huRu.

The remaining cases of the Kramer–Tunnell conjecture

For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$, motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some where $K$ is of characteristic $2$, and we complete its proof by reducing the positive characteristic case to characteristic $0$. For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_{p}$: we find an elliptic curve $E^{\prime }$ defined over a $p$-adic field such that all the terms in the Kramer–Tunnell formula for $E^{\prime }$ are equal to those for $E$.

UNITARY GRASSMANNIANS OF DIVISION ALGEBRAS

We consider a central division algebra over a separable quadratic extension of a base field endowed with a unitary involution and prove 2-incompressibility of certain varieties of isotropic right ideals of the algebra. The remaining related projective homogeneous varieties are shown to be 2-compressible in general. Together with [17], where a similar issue for orthogonal and symplectic involutions has been treated, the present paper completes the study of Grassmannians of isotropic right ideals of division algebras.

Real non-unital division algebras with [formula omitted] as derivation algebra

We obtain a family of non-unital eight-dimensional division algebras over a field F out of a separable quadratic field extension S of F, a three-dimensional anisotropic hermitian form over S of determinant one, and three invertible elements c,d,e∈S. These algebras contain a four-dimensional subalgebra which can be viewed as a generalization of a (nonassociative) quaternion algebra. The four-dimensional algebras are studied independently.Over R, this construction can be used to generate division algebras with derivation algebra isomorphic to su(3), which are the direct sum of two one-dimensional modules and a six-dimensional irreducible su(3)-module. Albert isotopes with derivation algebra isomorphic to su(3) are considered briefly as well.

Results on Witt kernels of quadratic forms for multi-quadratic extensions

In this paper we compute the Witt kernel of quadratic forms for the composition of a purely inseparable multi-quadratic extension with a separable quadratic extension. We also include the case of a multi-quadratic purely inseparable extension by completing the proof given before by the second author for such an extension.

Characterization of γ-Factors: the Asai Case

Let $E$ be a separable quadratic extension of a locally compact field $F$ of positive characteristic. Asai \gamma-factors are defined for smooth irreducible representations \pi of ${\rm GL}_n(E)$. If \sigma is the Weil-Deligne representation of $\mathcal{W}_E$ corresponding to \pi under the local Langlands correspondence, we show that the Asai \gamma-factor is the same as the Deligne-Langlands \gamma-factor of the Weil-Deligne representation of $\mathcal{W}_F$ obtained from \sigma under tensor induction. This is achieved by proving that Asai \gamma-factors are characterized by their local properties together with their role in global functional equations for $L$-functions. As an immediate application, we establish the stability property of \gamma-factors under twists by highly ramified characters.

Invariant subalgebras of involutorial quaternion division algebras

Let K/k be a separable quadratic field extension. For quaternion division algebras with K/k involutions τ, their τ-invariant k-subalgebras are studied. A complete description of such subalgebras up to k-isomorphisms is given. Bibliography: 6 titles.

Construction method for some real division algebras with su(3) as derivation algebra

We obtain a family of eight-dimensional unital division algebras over a field F out of a separable quadratic field extension S of F, a three-dimensional anisotropic hermitian form h over S of determinant one and an element c ∈ S× not contained in F. These algebras are not third-power associative.

Metabolic involutions

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.

Completing an extension of Tunnell's theorem

We look at a special case of a familiar problem: Given a locally compact group G, a subgroup H and a complex representation π + of G how does π + decompose on restriction to H. Here G is GL ( 2 , F ) + , where F is a nonarchimedian local field of characteristic not two, K a separable quadratic extension of F, GL ( 2 , F ) + the subgroup of index 2 in GL ( 2 , F ) consisting of those matrices whose determinant is in N K / F ( K ∗ ) , π + is an irreducible, admissible supercuspidal representation of GL ( 2 , F ) + and H = K ∗ under an embedding of K ∗ into GL ( 2 , F ) .

A Valuation Theory for Nonassociative Quaternion Algebras

A nonassociative quaternion algebra over a field F is a 4-dimensional F-algebra A whose nucleus is a separable quadratic extension field of F. We define the notion of a valuation ring for A, and we also define a value function on A with values from a totally ordered group. We determine the structure of the set on which the function assumes non-negative values and we prove that, given a valuation ring of A, there is a value function associated to it if and only if the valuation ring is integral and invariant under proper F-automorphisms of A.

Groups of outer type E6 with trivial Tits algebras

In two 1966 papers, J. Tits gave a construction of exceptional Lie algebras (hence implicitly exceptional algebraic groups) and a classification of possible indexes of simple algebraic groups. For the special case of his construction that gives groups of type E6, we connect the two papers by answering the question: Given an Albert algebra A and a separable quadratic field extension K, what is the index of the resulting algebraic group?

Involutions en degré au plus 4 et corps des fonctions d'une quadrique en caractéristique 2

The aim of this paper is to study in characteristic $2$ and degree at most $4$ the isotropy of involutions of the first kind and quadratic paires over the function field of a projective quadric. We also give a complete answer to the hyperbolicity over an inseparable quadratic extension in arbitrary degree. The case of a separable quadratic extension has been studied in [4].