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Publication# Anneau de Witt des G-formes et multiples de G-formes trace

Abstract

This thesis deals with the study of G-forms and particulary the trace form of a G-Galois algebra. Let k be a field of characteristic not two. Let G be a finite group and L a G-Galois algebra over k. We define the trace form qL by qL(x, y) = TrL/k(xy) for all x, y in L. This is a bilinear symmetric form which is G-invariant. In other words, qL is a G-form. We know that L has a self-dual normal basis if and only if the trace form qL is G-isomorphic to the unit G-form q0. This is an important reason to classify the trace forms. This work contains two different parts. In the first part, we study G-forms in general, putting a ring structure on their Witt group. We then proved an analogue of Pfister's theorem - i.e. there is no zero divisor of odd dimension - when k[G] is semi-simple, k is big enough and G is abelian. Counter-examples are given when these conditions are not fulfilled. In the second part of this thesis, we study the trace form qL of a G-Galois algebra. E. Bayer-Fluckiger and H. W. Lenstra proved that if G is of odd order, then qL is always G-isomorphic to the unit form. If G is of even order, this is no longer the case. However, if the field k is of cohomological 2-dimension less than or equal to 1, then E. Bayer-Fluckiger and J.-P. Serre gave a necessary and sufficient condition - in terms of cohomological invariants - for the trace form qL to be isomorphic to the unit form. M. Monsurrò generalized this result to fields of virtual cohomological 2-dimension equal to 1. However, in higher cohomological dimensions, it becomes very difficult to classify the trace form itself. But it is possible to give general results if we consider multiples of the trace form or more generally the product of the trace form by a quadratic form. E. Bayer-Fluckiger formulated 2 conjectures about the possibility of finding a complete system of invariants for such a product when the quadratic form lies in a certain ideal of the Witt ring of k depending on the cohomological dimension of the field. In this work, we prove the first conjecture for all cohomological dimensions and the second one for a field of cohomological 2-dimension equal to 2. A more general conjecture is proved including the fields of virtual cohomological 2-dimension equal to 2. Finally, the second conjecture of E. Bayer-Fluckiger is proved in all cohomological dimensions, but only when either the characteristic of k is non zero or the group G is abelian or a 2-group, or k is big enough.

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Quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K. If , and the quadratic form equals zero only when all variables are simultaneously zero, then it is a definite quadratic form; otherwise it is an isotropic quadratic form.

Lebesgue covering dimension

In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. For ordinary Euclidean spaces, the Lebesgue covering dimension is just the ordinary Euclidean dimension: zero for points, one for lines, two for planes, and so on. However, not all topological spaces have this kind of "obvious" dimension, and so a precise definition is needed in such cases.

Witt group

In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field. Fix a field k of characteristic not equal to two. All vector spaces will be assumed to be finite-dimensional. We say that two spaces equipped with symmetric bilinear forms are equivalent if one can be obtained from the other by adding a metabolic quadratic space, that is, zero or more copies of a hyperbolic plane, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector.

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Let K be a field with char(K) ≠ 2. The Witt-Grothendieck ring (K) and the Witt ring W (K) of K are both quotients of the group ring ℤ[𝓖(K)], where 𝓖(K) := K*/(K*)2 is the square