Integrated MSc-PhD Program


KSM3E03: Field and Galois Theory


Fields, The characteristic of a field, Extensions, Algebraic and transcendental elements, Transcendental numbers, Constructions with straight-edge and compass, Algebraically closed fields, Homomorphisms from simple extensions, Splitting fields, Homomorphisms of algebraic extensions, Multiplicity of roots, Separable polynomials, Perfect fields, Groups of automorphisms of fields, Separable, normal, and Galois extensions, The fundamental theorem of Galois theory, Constructible numbers, The Galois group of a polynomial, Solvability of equations, Polynomials of degree at most three, Quartic polynomials, Finite fields, Primitive element theorem, Fundamental Theorem of Algebra, Cyclotomic extensions, Dedekind’s theorem on the independence of characters, The normal basis theorem, Cyclic extensions, Kummer theory, Proof of Galois’s solvability theorem.

Suggested texts:

            1. Lang, Algebra
            2. Artin, E. Galois Theory
            3. Joseph Rotman, Galois Theory
            4. J.S. Milne, Fields and Galois Theory