MA 5070: Modules and Fields

All the updates related to MA 5070 are announced here.

### Class timings:

Monday: 2:30 pm to 04:00 pm

Thursday: 4:00 pm to 05:30 pm

### Examination and Grades:

The grades will be decided by the marks obtained in the assignments, mid-term, surprise tests, the final examination, and also by the performance in the class.

## Syllabus:

Review of Rings, Modules, Free modules, Cartesian products and direct sums of modules, quotient modules, Simple and semisimple modules, isomorphism theorems.
Modules over principal ideal domains and applications. Noetherian and Artinian rings/Modules, Hilbert basis theorem. Jordan-Holder theorem. Projective/Injective modules.

Field extensions. Algebraic/transcendental elements, Algebraic extensions. Finite fields, Cyclotomic fields. Splitting field of a polynomial. Algebraic closure of a field,
Uniqueness. Normal, separable, purely inseparable extensions. Primitive elements, simple extensions. Fundamental theorem of Galois theory. Solvability by radicals.
Solutions of cubic and quartic polynomials, Insolvability of quintic and higher degree polynomials. Geometric constructions.
## References:

- C. Musili, Introduction to Rings and Modules, Narosa Publishing House, New Delhi 1994.
- D. S. Dummit and R. M. Foote, Abstract Algebra, John Wiley & Sons Inc, 3rd Edition, 2004.
- S. Lang, Algebra, 3rd Edition, Springer-Verlag (India), 2004.
- Jacobson, N., Basic Algebra II, Hindustan Publishing Corporation, 1991.

## Assignments:

The assignments have to be submitted right after the class on the respective due dates. The assignments can be submitted after the due date and time as well,
but only 50% of the marks will be credited into your account, unless you have a valid reason for the late submission.