1. Solvable groups. Nilpotent groups.

2. Definition and examples of rings. Fields, subrings. homomorphism of rings. Kernel

and image of a homomorphism. Kernel is NOT a subring. Characteristic of a ring.

Quotient rings. Prime ideals, maximal ideals and their characterization. Polynomial

rings. Divisibility. units. Factorization in a ring. Irreducible and prime elements in a

ring. Unique factorization domain, principal ideal domain and euclidean domains.

3. Fields. Field extensions. Finite fields. Finite and algebraic extensions. Classical

geometric constructions. Galois theory- fundamental theorem of Galois theory and

Abel’s theorem.

**References**:

1. Joseph A Gallian, Contemporary abstract algebra, Narosa Publishers, India.

2. John B Fraleigh, A First Course in Abstract Algebra, Narosa Publishers, India.

3. M Artin, Algebra, Prentice Hall India.

4. Joseph Rotman, Galois Theory, Universitext, Springer.

- Teacher: Ramesh Venkadachalam

This course describes the methods to solve ODE and PDE with applications

- Teacher: Kokila J