MAT 240 - Algebra I

Syllabus for MAT 240 – Algebra I

This is a compulsory course for BS Mathematics students in their 3^rd semester.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Overview: Learning traditional Abstract Algebra in a contemporary style. The course will cover the standard algebraic structures of groups, rings and fields up to the Fundamental Theorem of Algebra.

Detailed Syllabus:

Module I: Groups

1. Definition and examples, abelian and non-abelian groups, finite and infinite groups
2. Subgroups: characterisations, subgroup generated by a subset, commutator subgroup, center
3. Cyclic Groups: Properties, classification of subgroups
4. Permutation Groups: definition and notation, examples, properties, Symmetric group on n letters (S_n), Alternating group (A_n) on n letters
5. Cosets and Lagrange's theorem
6. External Direct Product: Definition and examples, properties, criteria for external direct product to be cyclic, finitely generated abelian groups

Module II: Morphisms

7. Normal subgroups, factor groups, internal direct products
8. Group homomorphism: Definition and examples, properties
9. Isomorphism, First Isomorphism Theorem, automorphism, properties, examples

Module III: Rings

10. Introduction to Rings: Definition, examples, properties
11. Subrings
12. Ideals, factor rings, prime ideals and maximal ideals
13. Polynomial Rings: Notation and terminology, division algorithm

Module IV: Extension Fields

10. Integral Domain, definitions and examples, Fields, Characteristic
11. Examples of Fields, algebraic and transcendental elements, degree of a field extension
12. Finite Fields: examples, Fundamental Theorem of Algebra

Main Reference:

• Contemporary Abstract Algebra by Joseph A. Gallian, 4^th edition. Narosa, 1999.

Other References:

• Topics in Algebra by I.N. Herstein, 2^nd Edition. Wiley India, 2006.
• Algebra by Michael Artin, 2^nd Edition. Prentice Hall India, 2011.
• A First Course in Abstract Algebra by John B. Fraleigh, 7^th Edition. Pearson, 2003.
• Undergraduate Algebra by Serge Lang, 2^nd Edition. Springer India, 2009.