Course abstract

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 a light, 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 degree 2 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. 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 a clear 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 closure • Normal and Separable extensions • Solvability of equations. • Inverse Galois Problem