Integer quadratic forms and extensions of subsets of linearly independent roots

It is shown that quadratic extensions of a field not of characteristic two, which is linearly compact at a valuation, are determined by their groups of norms, provided the residue field has a unique quadratic extension and is perfect if of characteristic two. It is indicated that Henselian can replace linearly compact in some cases. Necessity of the condition on the residue field is shown. 1. In this brief paper we shall apply the techniques and results of the paper Quadratic extensions of linearly compact fields by Ron Brown and myself (referred to below as [BW]) to prove the following result: THEOREM 1. Let F be a field of characteristic char(F)$2. Let v be a (nonarchimedean) valuation on F with arbitrary value group rF and residue field kF; assume only that kF is perfect if char(kF)=2. Suppose that F is linearly compact at v and that kF has a unique quadratic extension. Then for K1 and K2 quadratic extensions of F, K1 K2 if and only if N1K1 = N2K2. (Here Ni denotes the norm map Ki-*F.) For definition and properties of linear compactness, see [BW] or [Bour]. All the hypotheses of Theorem 1 are satisfied by any classical local field of characteristic not two. Indeed Theorem 1 is a generalization of a special case of the local class field theorem which says that an abelian extension of a local field is determined by its group of norms. The conclusion of Theorem 1 is equivalent to the assertion that a binary quadratic form over F is determined up to equivalence by the elements of F which it represents. A straightforward application of the Global Squares Theorem extends this result to the global case, obtaining the well-known result that over any local or global field of characteristic not two, binary quadratic forms are equivalent if and only if they represent the same elements. Received by the editors July 9, 1971. AMS 1970 subject classifications. Primary 12B10, 12J10, 12B25; Secondary 10C05, 12A25, 12J20.

We develop the version of the $J$-invariant for hermitian forms over\nquadratic extensions in a similar way Alexander Vishik did it for quadratic\nforms. This discrete invariant contains informations about rationality of\nalgebraic cycles on the maximal unitary grassmannian associated with a hermitan\nform over a quadratic extension. The computation of the canonical $2$-dimension\nof this grassmannian in terms of the $J$-invariant is provided, as well as a\ncomplete motivic decomposition.\n

Let L be a unimodular Z-lattice on a quadratic space V over Q, dim ⁡ V ⩾ 3 \dim V \geqslant 3 , and let O \mathcal {O} be the ring of algebraic integers of the quadratic field E = Q ( m ) E = {\mathbf {Q}}(\sqrt m ) . We explicitly calculate the number of proper spinor genera in the genus of the lattice L ⊗ Z O L{ \otimes _{\mathbf {Z}}}\mathcal {O} .

In this paper, we analyze what happens with respect to quadratic forms when a square root is adjoined to a field F which has exactly two quaternion algebras. There are many such fields—the real numbers and finite extensions of the p-adic numbers being two familiar examples. For general quadratic extensions, there are many unanswered questions concerning the quadratic form structure, but for these special fields we can clear up most of them.It is assumed char F ≠ 2 and K = F (√a) where a ∊ Ḟ – Ḟ2. Ḟ denotes the non-zero elements of F. Generally the letters a, b, c, … and α, β, … refer to elements from Ḟ and x, y, z, … come from .

Pairs of quadratic forms over a quadratic field extension

This work consists of results on three questions in the algebraic theory of forms. The first question deals with characterizing the Witt kernel (i.e. the anisotropic non-singular quadratic forms over that become hyperbolic) over a given field extension. For separable quadratic and bi-quadratic extension this is well known (for example see (B1, 4.2 and 4.3), (B2, p. 121), (L, p. 200), (ELW, 2.12)). In chapter 2, we provide answers to this question for inseparable quadratic and bi-quadratic extensions. We provide theorem 2.1.5, which in particular answers question 4.4 in (B2). From this result we prove the excellence property for inseparable quadratic extension, which is in turn used to find the Witt kernels of inseparable bi-quadratic extensions. In the third chapter we study the relation between similarity of quadratic forms and isomorphism and place equivalence of their function fields. In sections 3.1 and 3.2, we show that the function fields of special Pfister neighbors of the same Pfister form are isomorphic. Also we show that any Pfister neighbor of codimension $\\le$4 is special; in particular this implies place equivalence implies birational equivalence in this case. Together with the main result of (H3), this gives an affirmative answer of the quadratic Zariski problem in dimension 3. (see S 3.3). In S 3.4 we provide few results on the problem of descent of similarity over field extensions and some examples where similarity is determined by their generic splitting tower. In the last chapter we provide a positive answer for the following conjecture of Pfister-Leep in the special case d = the characteristic of the field k C scONJECTURE. For a fixed d, if k is a $C\\sbsp{0}{d}$-field, then k is a p-field for some prime $p \\ne d.$.

For various types of field extensions [Formula: see text] and values of [Formula: see text] we consider quadratic forms [Formula: see text] that lie in the [Formula: see text]th power [Formula: see text] of the fundamental ideal of [Formula: see text] but are already defined over [Formula: see text]. We then search for some [Formula: see text] of minimal dimension that maps to [Formula: see text] when extending the scalars to [Formula: see text]. This problem can be easily solved completely for finite extensions of odd degree. For quadratic extensions [Formula: see text], the situation is more involved, but solved for [Formula: see text]. For [Formula: see text], we construct quadratic field extensions [Formula: see text] and a form [Formula: see text] such that any form [Formula: see text] that maps to [Formula: see text] when extending the scalars to [Formula: see text] has dimension at least [Formula: see text].

If K/F is a quadratic extension, we give necessary and sufficient conditions in terms of the discriminant (resp. the Clifford algebra) for a quadratic form of dimension 2 (resp. 4) over K to be similar to a form over F. We give similar criteria for an orthogonal involution over a central simple algebra A of degree 2 (resp. 4) over K to be such that A = A ′ ⊗ F K A = A’ { \otimes _F}K , where A ′ A’ is invariant under the involution. This leads us to an example of a quadratic form over K which is not similar to a form over F but such that the corresponding involution comes from an involution defined over F.

Similarity of quadratic forms and related problems

Introduction Classical theory of symmetric bilinear forms and quadratic forms: Bilinear forms Quadratic forms Forms over rational function fields Function fields of quadrics Bilinear and quadratic forms and algebraic extensions $u$-invariants Applications of the Milnor conjecture On the norm residue homomorphism of degree two Algebraic cycles: Homology and cohomology Chow groups Steenrod operations Category of Chow motives Quadratic forms and algebraic cycles: Cycles on powers of quadrics The Izhboldin dimension Application of Steenrod operations The variety of maximal totally isotropic subspaces Motives of quadrics Appendices Bibliography Notation Terminology.

Two different proofs are given showing that a quaternion algebra Q defined over a quadratic etale extension K of a given field has a corestriction that is not a division algebra if and only if Q contains a quadratic algebra that is linearly disjoint from K. This is known in the case of a quadratic field extension in characteristic different from two. In the case where K is split, the statement recovers a well-known result on biquaternion algebras due to Albert and Draxl.

The Kaplansky radical of a quadratic field extension

ABSTRACTFrom a normed quadratic space (V,q), we construct a norm on the Clifford algebra C(V,q). We describe the associated graded form of this norm and give a condition for this norm to be a gauge. Then, we apply our results to prove that for a complete discrete valued field, an anisotropic quadratic form q with dimq = 0 mod 8 and nonsplit Clifford algebra cannot be at the same time a transfer of a K-hermitian form with K∕F an inertial quadratic field extension and a transfer of a T-hermitian form with T∕F a ramified quadratic field extension.

is bijective for any field F of characteristic two and for any n>0. Here, K,(F) is the K-group of Milnor defined in [11] and I is the unique maximal ideal of the Witt ring W(F) of non-degenerate symmetric bilinear forms over F. On the other hand, Sah studied quadratic forms in characteristic two in [15]. Let F be a field of characteristic two, Wq(F) the Witt group of non-degenerate quadratic forms over F, and define I'Wq(F) (n>O) by the canonical W(F)-module structure on Wq(F). Then, he proved that Wq(F)/IWq(F) is isomorphic to the group of quadratic cyclic extensions of F and IWq(F)/I2Wq(F) is isomorphic to the subgroup Br(F)2 of the Brauer group Br(F) of F consisting of elements annihilated by 2. In this paper, we prove the above Milnor's conjecture, and describe the precise structures of W(F) and Wq(F) generalizing the result of Sah. For any field F of characteristic p>0, let ~ be the n-th exterior power over F of the absolute differential module n t _g21 Let d(~2~ 1) be the .~F/Z-F/FP" image of the exterior derivation d: f2~ t --~ f2~, and let v(n)v (resp. H"~+1(F)) be the kernel (resp. cokernel) of the homomorphism

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.