Galois Theory

Review of Groups : Groups actions, Composition series, Jordan Holder theorem, Solvable groups.

Review of Rings : Polynomial rings, Zeros of polynomials, Elementary symmetric functions and
Fundamental Theorem on Symmetric Functions. Resultants and Discriminants, Euclidean rings,
Principal ideal domains and Factorial rings, Factorization in polynomial rings.

Field theory : Finite Fields, Finite and Algebraic extensions. Algebraic closure, Algebraically
closed fields. Proof of Fundamental Theorem of Algebra. Separable polynomials and Separable
extensions. Splitting fields, Normal extensions, Galois extensions, Galois group of a polynomial.
Fundamental Theorem of Galois theory, Radical extensions, Solvability by radicals, Computation
of Galois groups.

Books

• Artin, E., Galois Theory, University of Notre Dame Press, 1944
• Artin, M., Algebra, Prentice-Hall, 1994.
• Jacobson, N., Lectures in Abstract Algebra, Vols. I, II & III, D. Van Nostrand Co. Inc., Princeton, New Jersey, 1966.
• Lang, S., Algebra, Graduate Texts in Mathematics, Vol. 211, Springer-Verlag, 2002
• Weber, H., Lehrbuch der Algebra, Band I, II, III, Braunschweug 1898, 1899, 1908.