MTH 410 (Field Theory and Galois Theory)
Instructor : Dr. Anupam Kumar Singh
Audience : BS-MS students at IISER Pune
Schedule :
August - November 2013, 3 hours (2 lectures + 1 tutorial) a week
Monday, Tuesday and Thursday at 13:30
Evaluation :
Test I - 25%
Test II (mid sem exam) - 25%
Mini Project - 25%
Test III (end sem exam) -25%
Prerequisite : Basic knowledge of groups and linear algebra.
Goal of the Course : Galois theory is one of the crowning glories that paved the way to modern algebra. The subject originated from the question of solving polynomial equations over a field. Abel and Galois were main contributor to the subject in begin. The concept of Groups origintaed from Galois' work and the subject is absolutely essential to understand Algebraic Number Theory, Commutative Algebra and most of the algebra in modern times.
Proposed Course Content :
Field Extensions: Finite, algebraic and transcendental extensions, adjunction of roots, degree of a finite extension, algebraically closed fields, existence and uniqueness of algebraic closure, splitting fields, normal extensions, separable extensions, Galois extensions, automorphism groups and fixed fields, fundamental theorem of Galois theory, examples: finite fields, cyclic extensions, cyclotomic extensions, solvability by radicals, ruler and compass constructions, constructibility of regular n-gon.
Text Books :
Abstract Algebra: Dummit and Foote, Wiley India.
Galois Theory (lectures delivered at the University of Notre Dame): Emil Artin, Dover
Field Theory: Roman, Springer.
Galois’ Theory of Algebraic Equations: J.-P. Tignol, World Scientific.