NOTE: Attempt any FIVE questions selecting at least TWO questions from each section.

Section- I   (5/9)

Congruences

•    Elementary properties of prime numbers

•    Residue classes and Euler’s function

•    Linear congruences and congruences of higher degree

•    Congruences with prime moduli

•    The theorems of Fermat, Euler and Wilson

Number-Theoretic Functions

•    Möbius function

•    The function [x], The symbols O and their basic properties

Primitive roots and indices

•    Integers belonging to a given exponent  (mod p)

•    Primitive roots and composite moduli

•    Determination of integers having primitive roots

•    Indices, Solutions of Higher Congruences by Indices

Diophantine Equations

•    Equations and Fermat’s conjecture for n = 2, n = 4



Section-II (4/9)

Quadratic Residues

•    Composite moduli, Legendre symbol

•    Law of quadratic reciprocity

•    The Jacobi symbol

Algebraic Number Theory

•    Polynomials over a field

•    Divisibility properties of polynomials

•    Gauss’s lemma

•    The Eisenstein’s irreducibility criterion

•    Symmetric polynomials

•    Extensions of a field

•    Algebraic and transcendental numbers

•    Bases and finite extensions, Properties of finite extensions

•    Conjugates and discriminants

•    Algebraic integers in a quadratic field, Integral bases

•    Units and primes in  a quadratic field

•    Ideals, Arithmetic of ideals in an algebraic number field

•    The norm of an ideal, Prime ideals