Algebra-III

Course Code: MA 407; MM 505 | Credits: 4 | Prerequisite: Algebra-I, Algebra-II

Unit I: (Field extensions) Mostly field extension, algebraic extensions, the degree of an extension. Many examples will be discussed for a better understanding of these concepts. Some impossibility theorem in the Ruler and compass constructions proved as an application to algebraic extensions.
Unit II: (More field extensions) The idea behind Galois theory, Normality and separability of extensions, Field degrees, and group orders, normal closures. Discuss many examples and counterexamples for a better understanding of these ideas.
Unit III: (The Galois correspondence) Galois correspondence, explain a close relationship between two different subjects group theory and field extensions along with examples.
Unit IV: (Applications of Galois Theory) Applications of the Galois correspondence to the diverse areas, i.e., solution of equations by radicals, the general polynomial equation, Finite fields, regular polygons, the Fundamental Theorem of Algebra.
Unit V: (Transcendental extensions) Transcendental extensions, Luroth’s theorem.
Unit VI: (Commutative Algebra) Concepts from commutative algebra: Simple Modules, Jordon-Holder theorem for Modules, Jacobian, and Nilradicals, Nakayama Lemma, Chinese remainder theorem, tensor products. The Artinian ring implies Noetherian.

Define various notions from field extensions, for example algebraic, transcendental, normal separable, the degree of an extension, etc.
Identify and apply the above notions to various examples.
State and understand the Galois correspondence.
Analyze some examples.
Understand how to apply the Galois correspondence to other areas in mathematics, like classical or algebraic geometry, or in commutative algebra.
Define various concepts from commutative algebra, such as Jordan-Holder series, Noetherian ring, Artinian ring. A student should understand theorems like Nakayama lemma, Chinese remainder theorem.

Michael Artin, Algebra, 2nd ed. Pearson, Upper Saddle River, NJ, 2011.
Herstein, I.N., Topics in Algebra, 3rd ed. Wiley, New York, 1996.
Dummit and Foote, Abstract Algebra, 3rd ed. Wiley, New York, 2003.
Lang, Serge, Algebra, revised third edition, Graduate Texts in Mathematics, 211, Springer-Verlag, New York, 2002.
Jacobson, Nathan, Basic Algebra, Volume 1, second edition, W. H. Freeman and Company, New York, 1985.
Jacobson, Nathan, Basic Algebra, Volume 2, second edition, W. H. Freeman and Company, New York, 1989.
Jacobson, Nathan, Lectures in Abstract Algebra, III, Theory of fields and Galois theory, Second corrected printing, Graduate Texts in Mathematics, No. 32. Springer-Verlag, New York-Heidelberg, 1975.