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.

"Happy Abstract Algebra Classes"

This article by John Fraleigh on how he revamped the evaluation processes for his Abstract Algebra classes is quite interesting. His book on Abstract Algebra first made me happy about the subject, so I am certainly biased towards accepting his claims...