On successful completion of this module, students will be able to:
Apply the Euclidean algorithm to integers and polynomials;
Use modular arithmetic, apply the Chinese Remainder Theorem to systems of simultaneous congruences;
Apply the Quadratic Reciprocity Law to compute various Legendre symbols and to solve quadratic congruences;
State and prove functional equation for the Riemann zeta function and formulate the Riemann hypothesis;
Prove that there are infinitely many primes in an arithmetic progression;
Demonstrate that a given polynominal is irreducible over integers;
Define norm and trace maps for algebraic field extensions, give examples of separable and inseparable extensions;
Formulate and apply basic statements of Galois theoory in the contexts of algebraic numbers and finite fields;
Apply Hensel's lemma to solve polynominal congruences modulo prime powers
The goal of this course is to introduce the students to most of the basic concepts of number theory, both analytic and algebraic. We begin with the classical questions such as factorization, distribution of primes annd finding integer solutions to algebraic equations. Second part of the course deals with analytic methods. We will study Riemann zeta function, discuss Riemann hypothesis and prove the theorem on primes in arithmetic progressions. Third part will be devoted to algebraic field extensions and Galois theory. We will study algebraic numbers, finite fields and p-adic fields, and see that the theorem on primes in arithmetic progressions follows from Chebotarev's density theorem. The course will be accompanied by bi-weekly tutorials in the form of problem-solving sessions. Recommended reading consists of (selected chapters from) books (1,2,3,4) below.
Some classical Diophantine equations. Pythagorean triples.
Modular arithmetic. Fermat's little theorem. Euler's theorem. Chinese Remainder Theorem. Quadratic reciprocity law.
Analytic continuation an dfunctional equation for Riemann zeta function. Riemann Hypothesis. Distribution of primes.
Dirichlet characters. L-functions. Primes in Arithmetic Progressions.
Algebraic numbers. Galois theory.
Algebraic integers. Finiteness of class numbers. Algebraic units and Dirichlet's theorem.