### MA 212: Algebra I

#### Part A: Group theory

1. Basic definitions, examples
2. Cyclic groups and its subgroups
3. Homomorphisms, quotient groups, isomorphism theorems
4. Group actions, Sylow’s theorems, simplicity of $A_n$ for $n\geq 5$
5. Direct and semi-direct products
6. Solvable and nilpotent groups
7. Free groups

#### Part B: Ring theory

1. Basic definitions, examples
2. Ring homomorphisms, quotient rings, properties of ideals
3. Localization, ring of fractions
4. The Chinese remainder theorem
5. Euclidean domains, principal ideal domains, unique factorization domains
6. Polynomial rings over fields, irreducibility criteria

#### Part C: Module theory

1. Basic definitions and examples
2. Homomorphisms and quotient modules
3. Direct sums and free modules
4. Tensor product of modules
5. Structure theorem of modules over PID’s and consequences
6. Noetherian rings and modules, Hilbert basis theorem

#### Suggested books and references:

1. Artin, Algebra, M. Prentice-Hall of India, 1994.
2. Dummit, D. S. and Foote, R. M., Abstract Algebra, McGraw-Hill, 1986.
3. Lang, S., Algebra (3rd Ed.), Springer, 2002.
4. Hungerford, Algebra, Graduate Texts in Mathematics 73, Springer Verlag, 1974.
5. Nathan Jacobson, Basic Algebra I & II, Dover, 2009.
6. Nathan Jacobson, Lectures in Abstract Algebra I, II & III, Graduate Text in Mathematics, Springer Verlag, 1951.

#### All Courses

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 20 Mar 2023