Aims & Learning Objectives:
Aims: This course develops the basic theory of rings and fields and expounds the fundamental theory of Galois on solvability of polynomials.
At the end of the course, students will be conversant with the algebraic structures associated to rings and fields. Moreover, they will be able to state and prove the main theorems of Galois Theory as well as compute the Galois group of simple polynomials.
Rings, integral domains and fields.
Field of quotients of an integral domain. Ideals and quotient rings. Rings of polynomials. Division algorithm and unique factorisation of polynomials over a field.
Extension fields. Algebraic closure. Splitting fields. Normal field extensions. Galois groups. The Galois correspondence.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN EVEN YEAR.