Groups, fields, Galois theory, and applications

Abstract

In tis project we shall study some topics of significant relevance for Contemporary Algebra. First we shall study some basic notions of Group Theory, mainly finite groups. Afterwards we approach elements of Field Theory and extensions, using the language of polynomials and roots. We shall cover finite, algebraic, f.g. extensions. Then we pass to symmetric polynomials and roots, and their connections to field extensions. Then we pass to detailed study of field extensions: normal and separable extensions. Afterwards we pass to the fundamentals of Galois theory, relating extensions to automorphism groups. We shall study soluble groups and as a consequence, we shall deduce the theory of ruler and compass constructions, including the three classical problems, and the Gauss theorem about the regular polygons. These are important topics in Algebra (and in Mathematics in general) since Galois theory shows us how to relate two quite distant theories (groups and fields) in order to solve extremely important problems like solubility of equations and RC constructions. These are topics in Algebras that enrich the students math culture and give a solid base for future research. These are fundamental theories lacking their knowledge one cannot think of any serious research in Algebra or Number theory. We chose a generic theme that serves as a basis for various topics. The IC is planned for 12 months. We took into account Lucas is a freshman in Maths, and will be taking serious theoretical courses like Analysis, Linear algebra, Groups, Rings and Fields, Topologuy of Metric spaces. We took into account that the student, having done several such courses, could decide better which area of Maths he likes more. The project admits various natural developments after the first year, depending on Lucas interests. One such could be the study of central simple algebras and division rings, coming to Brauer groups, and afterwards, algebraic properties of division rings, up to the famous theorem of Amitsur on crossed products. Suc stidues require notions from polynomial identities and generic matrices. It is a topic near the supervisors research interests. Another topic that continues the study described above might be algebraic number theory. But the choice of the topics for future research will depend on the student and his affinity with some of the topics described (or even something different). The student has some basic notions in Algebra. He already studied some topics in Linear algebra and Group theory, as an IC without a grant, during the first semester of 2019. Hence we believe that he will absorb quickly the algebraic notions necessary for developing the project.