Algebra-I

Algebra - I (Ph.D. and iPh.D. course)

August-November 2016

Instructor : Dr. Anupam Singh

Schedule : Lectures on Tuesday, Wednesday 3pm-4pm

Tutorial on Thursday 3pm-4pm

Evaluation : mid-semester exam 25 %

end-semester exam 25 %

tests 50 %

Prerequisite : enthusiasm towards mathematics

Goal of the course : This course is a compulsory part of course work to get PhD degree at IISER Pune. The course aims to revise basic algebra learned during under graduate degree programme and introduces topics which will be useful towards research in mathematics.

Proposed course content :

Group Theory

Groups, subgroups, cyclic subgroups, normal subgroups and quotient groups, Lagrange's theorem, homomorphisms and isomorphism theorems.
Cauchy's theorem, composition series, Jordan-Holder theorem.
Group action and permutation representation, centralizer and normalizers, stabilizers and kernels, Cayley's theorem and the class equation.
Automorphisms, inner automorphisms, automorphism groups of some finite groups.
Simple groups, alternating groups and their simplicity for n > 4. Sylow theorems and applications.

Rings and Fields

Rings, examples of polynomial rings, matrix rings, group rings, ring homomorphism, quotient rings and isomorphism theorems, ideals and their properties, prime and maximal ideals, existence of maximal ideals.
Chinese remainder theorem, Euclidean domains, PIDs, UFDs, field of fractions, characteristic, finite fields, Eisenstein's irreducibility criterion, Gauss’ theorem.
Non-commutative rings: semi-simple rings and modules. Wedderburn's theorem.
Fields, fields of characteristic p > 0, algebraic field extension, algebraic closure, perfect field, separable and inseparable extensions, primitive element theorem, extensions of finite fields.
Cyclotomic polynomials, Galois theory (fundamental theorem), solvability by radicals, Galois groups of polynomials, Galois groups over rationals, transcendental extensions.

Linear Algebra :

Vector Spaces, linear transformation, basis, dimension, matrices, dual vector spaces, modules, examples of Z-modules and F[X]-modules, characteristic and minimal polynomial, eigenvalues, eigenvectors, Cayley-Hamilton theorem.
Inner product spaces, bilinear forms, symmetric, skew-symmetric forms, orthogonal bases, Gram-Schmidt orthogonalization process, Sylvester’s theorem, definition of orthogonal and symplectic groups.
Spectral theorem.
Modules over PIDs, diagonalization, triangulation, canonical forms: rational and Jordan form.
Tensor product of modules.

Text books and references :