### MA 414   Algebra I

• #### Course Timings:

Tue   11.35 AM - 12.30 AM,   Wed, Fri   11.00 AM - 12.30 PM.

The official time slots are 3b, 6a, 6b and the Exam slot is 6. We may follow a revised schedule in the first two weeks due to the CIMPA Workshop and Conference on Commutative Algebra. The revised timings will be finalized during the first class on Wednessday, Jan 2.

• #### Venue:

Room 113, Department of Mathematics
• #### Office Hours:

Tuesdays, 4.30 - 5.30 PM (Room 106 B, Maths Dept)

• #### Prerequistes:

MA 401 Linear Algebra,   MA 405 Basic Algebra

• #### Syllabus :

Simple groups and solvable groups, nilpotent groups, simplicity of alternating groups, composition series, Jordan-Holder Theorem. Semidirect products. Free groups, free abelian groups.

Rings, Examples (including polynomial rings, formal power series rings, matrix rings and group rings), ideals, prime and maximal ideals, rings of fractions, Chinese Remainder Theorem for pairwise comaximal ideals.

Euclidean Domains, Principal Ideal Domains and Unique Factorizations Domains. Polynomial rings over UFD's.

Fields, Characteristic and prime subfields, Field extensions, Finite, algebraic and finitely generated field extensions, Classical ruler and compass constructions, Splitting fields and normal extensions, algebraic closures. Finite fields, Cyclotomic fields, Separable and inseparable extensions.

Galois groups, Fundamental Theorem of Galois Theory, Composite extensions, Examples (including cyclotomic extensions and extensions of finite fields).

Norm, trace and discriminant.

Solvability by radicals, Galois' Theorem on solvability.

Cyclic extensions, Abelian extensions, Transcendental extensions.

• #### Text/References:

M. Artin, Algebra, Prentice Hall of India, 1994.

D.S. Dummit and R. M. Foote, Abstract Algebra, 2nd Ed., John Wiley, 2002.

J.A. Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa, 1999.

N. Jacobson, Basic Algebra I, 2nd Ed., Hindustan Publishing Co., 1984, W.H. Freeman, 1985.