MA 5430 ALGEBRA II: RING THEORY AND FIELD THEORY


Course Details

Module Theory: Right and Left modules, Examples over Z, polynomial rings, Quotient modules, module homomorphism & homomorphism theorems .
Ring Theory: Structure Theorem of modules over PID, a polynomial ring over a UFD is again a UFD, basic properties of Noetherian rings and modules, basic properties of localisation .
Field Theory: Classical Straightedge and Compass construction and examples, normal, separable and Galois extensions.
Basic Non-Commutative Algebra: Linear Maps and Modules over non-commutative rings, Simple and Semi-simple modules and rings, Schur's Lemma, Jacobson's Density Theorem, Burnside Lemma and Wedderbern Theorem, Matrix rings over division algebras are simple.

Course References:

Text Books:
1. D. S. Dummit and R. M. Foote: Abstract Algebra, 2nd Edition, John-Wiley, 1999.
2. S. Lang: Algebra 3rd Edition, Addison-Wesley, 1999.
Reference:
1. J.A. Gallian: Contemporary Abstract Algebra, 4th Ed., Narosa, 1999.
2. M. Artin: Algebra, Prentice Hall inc 1994.
3. I.N. Herstein: Topics in Algebra, John-Wiley, 1995.
4. T. A. Hungerford: Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, 1980.