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# Introduction to Galois Theory

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### Overview

In this introductory course on Galois theory, we will first review basic concepts from rings and fields, such as polynomial rings, field extensions and splitting fields. We will then learn about normal and separable extensions before defining Galois extensions. We will see a lot of examples and constructions of Galois groups and Galois extensions. We will then prove the fundamental theorem of Galois theory which gives a correspondence between subgroups of the Galois group and intermediate fields of a Galois extension. We will then cover some important applications of Galois theory, such as insolvability of quintics, Kummer extensions, cyclotomic extensions. This course will focus a lot on solving exercises and giving plenty of examples. We will give several exercises to be done by students and will have weekly problem solving sessions where we will solve problems in detail. INTENDED AUDIENCE :Final year B.Sc students or M.Sc students in mathematics.PREREQUISITES : Courses in linear algebra, group theory, rings and fields are prerequisites.INDUSTRIES SUPPORT :None

### Syllabus

Week 1:Review of rings and fields I: polynomial rings, irreducibility criteria, algebraic elements, field extensions Week 2:Review of rings and fields II: finite fields, splitting fieldsWeek 3:Normal extensions, separable extensionsWeek 4:Fixed fields, Galois groups
Week 5:Galois extensions, properties and examplesWeek 6:Fundamental theorem of Galois theoryWeek 7:Solvability by radicals, insolvability of quinticsWeek 8:Kummer extensions, abelian extensions, cyclotomic extensions

### Taught by

Prof. Krishna Hanumanthu

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