Rings, Fields and Galois Theory

• Grant number: 24/23034-2
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: March 01, 2025
• End date: February 28, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Plamen Emilov Kochloukov
• Grantee: João Rafael Silva de Azeredo
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

In this project we intend to study the basics of the theory of com-mutative rings and fields, aiming at Galois theory. We start with the basic properties of commutative rings, such as Euclidean, principal,and factorial rings. We shall study the structure of polynomial ringsin one and several variables over a field (and over a domain). We shallreview the notion of symmetric polynomials, and Newton's formulas.Next we will move on to roots of polynomials, and the basic factsin this direction. We will follow with a brief introduction to grouptheory, with an emphasis on finite groups and permutations. We willintroduce the notion of soluble groups and simple groups. We willstudy the symmetric groups Sn, describe the subgroups of S4, andshow the simplicity of An, n not equal to 4. Next, we study the existence of roots of a polynomial, the notion of a splitting field, its existence and uniqueness. We will study field extensions: finite, algebraic, finitely generated. We will show the existence of an algebraic closure of a field. We consider the notions of separable and purely inseparableextension, normal extension. We will see the basics of the theory offinite fields and their automorphisms. Having seen all this, we studyGalois theory in general. In addition to the fundamental theorem ofGalois theory, we will see the notions of solubility in radicals, relatingthem to soluble extensions, abelian extensions and cyclic extensions.We will see examples of equations of degree greater or equal to 5 that cannot be solved in radicals. We will study an algorithm to determine the Galois group of a polynomial of degree 4 over the rationals. We shall also cover the ruler and compass constructions.The project will be based on the books by S. Lang [3], and by E.Artin [1].