Galois Theory, profinite groups and applications in quadratic forms

• Grant number: 14/08234-3
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Effective date (Start): June 01, 2014
• Effective date (End): May 31, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Hugo Luiz Mariano
• Grantee: Kaique Matias de Andrade Roberto
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

In the context of the infinite Galois theory over a base field, some questions are well answered having available an infinite extension of this, for example, the absolute Galois group of a field $F$, $Gal(F^s|F)$, where $F^s$ denotes a separable closure of $F$.However, in that broad setting, we must pay the price of having to introduce topological notions (the bijective correspondence is between subextensions and closed subgroups. Thus, we will develop the topological and categorical framework needed for this. In developing the theory, we prove that the groups of automorphisms of arbitrary Galois extensions are profinite, ie are projective limits of finite and discrete groups, thus inheriting a Boolean topology compatible with the group operations.In the detection process of the properties of absolute Galois groups we will develop studies on galoisian cohomology and profinite cohomology in general. A case of particular interest to the Algebraic Theory of Quadratic Forms (ATQF) is the "graded ring of cohomology of a field $F$", $H(F): =H(Gal(F^s|F); \pm 1)$.Thus we develop applications in the study of ATQF, relating the graded Witt ring of $F$ (obtained from the Witt ring of $F$ which classifies its regular and not isotropic quadratic forms) with the graded ring cohomology of $F$ and the graded ring reduced Milnor k- theory. In the sequence, we study the three graded rings in the context of the Milnor conjecture (demonstrated in the late 1990s), which identifies the three rings, anduse this to solve other questions about graded Witt ring.