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separable extensions, automorphism group and fixed fields fundamental theorems of
Galois theory and algebra.

THEORY (45 Hours)

UNIT I Field Extensions (15 Hours)
Irreducible polynomials and Eisenstein criterion, Adjunction of roots, Algebraic extensions,
algebraically closed fields, Splitting fields, Normal extensions, Multiple roots.
UNIT II Finite Fields (15 Hours)

Prime Fields, Finite fields, Roots of Irreducible Polynomials, Roots of unity and cyclotomic
polynomials, Representation of Elements of Finite Fields, Order of Polynomials and Primitive
Polynomials, Irreducible Polynomials.

UNIT III Galois Theory and its Applications (15 Hours)
Separable extensions, Automorphism groups and fixed fields, Fundamental theorem of Galois
theory, Fundamental theorem of algebra.

*TUTORIAL (15 Hours (1 Hour per week))

SUGGESTED READING:

1. P.B. Bhattacharya, S.K. Jain & S.R. Nagpaul, ‗Basic Abstract Algebra‘, Second Edition,
Cambridge University Press.
2. Rudolf Lidl & Harald Niederreiter, ―Finite Fields‖, Cambridge University Press.

Math.422 Partial Differential Equations 3+1*

LEARNING OBJECTIVES:

The primary objective of this course is to introduce:
 The Basic concepts related to partial Differential equations of first order and various
methods to solve these equations.
 The classification of second order partial differential equations, their canonical forms and
concept of adjoint operators. Derivation of Laplace equation/Poisson equation/ heat
equation/wave equations from basic concepts and their basic properties.
 The Laplace equation (elliptic equation), Heat equation (Parabolic equation) and Wave
equation (hyperbolic equation) by variable separable method and solve some boundary
value problems by some standard methods.
 The Laplace, heat and Wave equations in various coordinate systems and solve them.
Learn the use of theory and solutions/tools in solving the dynamical problems arising in
engineering and physical sciences

LEARNING OUTCOMES:

After the completion of the course, students will be able to
 Understand the Basic concepts related to partial Differential equations of first order and
various methods to solve these equations.
 Understand the classification of second order partial differential equations, their canonical
forms and concept of adjoint operators.
 Derivation of Laplace equation/Poisson equation/ heat equation/wave equations from basic
concepts and their basic properties.

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