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# Galois Theory

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## Overview

Galois Theory is showpiece of a mathematical unification which brings together several different branches of the subject and creating a powerful machine for the study problems of considerable historical and mathematical importance. This course is an attempt to present the theory in such alight, and in a manner suitable for undergraduate and graduate students as well as researchers.This course will begin at the beginning. The quadratic formula for solving polynomials of degree2 has been known for centuries and is still an important part of mathematics education. The corresponding formulas for solving polynomials of degrees 3 and 4 are less familiar. These expressions are more complicated than their quadratic counterpart, but the fact that they exist comes as no surprise. It is therefore altogether unexpected that no such formulas are available for solving polynomials of degree ≥ 5. A complete answer to this intriguing problem is provided by Galois theory. In fact Galois theory was created precisely to address this and related questions about polynomials. This feature might not be apparent from a survey of current textbooks on university level algebra.This course develops Galois theory from historical perspective and I have taken opportunity to weave historical comments into lectures where appropriate. It provides a platform for the development of classical as well as modern core curriculum of Galois theory. Classical results by Abel, Gauss,Kronecker, Legrange, Ruffini and Galois are presented and motivation leading to a modern treatment of Galois theory. The celebrated criterion due to Galois for the solvability of polynomials by radicals.The power of Galois theory as both a theoretical and computational tool is illustrated by a study of the solvability of polynomials of prime degree.The participant is expected to have a basic knowledge of linear algebra, but other that the course is largely self-contained. Most of what is needed from fields and elementary theory polynomials is presented in the early lectures and much of the necessary group theory is also presented on the way.Classical notions, statements and their proofs are provided in modern set-up. Numerous examples are given to illustrate abstract notions. These examples are sort of an airport beacon, shining aclear light at our destination as we navigate a course through the mathematical skies to get there.Formally we cover the following topics :Galois extensions and Fundamental theorem of Galois Theory.Finite Fields, Cyclic Groups, Roots of Unity, Cyclotomic Fields.Splitting fields, Algebraic closureNormal and Separable extensionsSolvability of equations.Inverse Galois Problem

## Syllabus

Week 1  :  Prime Factorisation in Polynomial Rings, Gauss’s Theorem
Week 2  :  Algebraic Extensions
Week 3  :  Group Actions
Week 4  :  Galois Extensions
Week 5  :  Finite Fields, Cyclic Groups, Roots of Unity, Cyclotomic Fields
Week 6  :  Splitting Fields, Algebraic Closure
Week 7  :  Normal and Separable Extensions
Week 8  :  Norms and Trace
Week 9  :  Fundamental Theorem on Symmetric
Week 10  :  Proof of the Fundamental Theorem Polynomial, of Algebra
Week 11  :  Orbits of the action of Galois group
Week 12  :  Inverse Galois Problem

#### Taught by

Prof. Dilip Patil

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