On successful completion of this module, students will be able to:
State and explain relationships between properties of field extensions and properties of their automorphism groups.
Construct explicitly finite fields of low orders.
Determine Galois groups of polynomials of low degree
Illustrate on specific examples applications of Galois theory.
Galois theory demonstrates how to use symmetries of objects to learn something new about properties of those objects, on the example of polynomial equations in one variable with coefficients in a field, and specifically roots of those equations. Students taking the module will see how basics of group theory can be used for solving problems outside group theory, in particular for proving a celebrated result of Abel on non-existence of formulas to solve equations of degree 5 using only arithmetic operations and extracting roots.
Recollection of relevant results in on groups, fields, and rings. Polynomial rings: UFD/PID property, Gauss lemma, Einstein's criterion.
Algebraic field extensions. Tower Law, ruler and compass constructions.
Splitting fields, and their properties. Classification of finite fields.
Normal and separable extensions. The Primitive Element Theorem. Galois extensions. The Galois correspondence. The Fundamental Theorem of Algebra via Galois Theory
Algorithm for computing the Galois group of a given polynomial.
Solubility by radicals. Cyclic, Abelian, solvable field extensions. Abel theorem on equations of degree five.
Abelian and cyclotomic extensions. Towards Kronecker-Weber Theorem
This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 20% to the final grade for the module at the annual examination session. Supplemental exams if required will consist of 100% exam.