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.

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.

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