Topics in algebraic number theory and Diophantine approximation
Date: (12th – 17th ) March, 2017
Place: Mathematics Department, College of Science, Salahaddin University/Erbil-Kurdistan Region/IRAQ.
Lecturers
Emeritus Professor, Universit´e P. et M. Curie (Paris VI)
Pierre Cartier
Professor of Algebra
full Professor of Algebra, Università Roma Tre, Italy
Professore a contratto,Università Roma Tre, Italy
Local Organizer Coordinator
Kawa M. A. MANMI
Head of the Mathematics Department, College of Science,
Salahaddin University/Erbil-Kurdistan Region/IRAQ
Scientific Committee
1. Herish O. Abdullah, The Dean of College of Sciences, department of Mathematics,
Salahaddin University/Erbil-Kurdistan Region/ – IRAQ
2. Kawa M. A. Manmi, Head of the Mathematics Department,
College of Science, Salahaddin University, /Erbil-Kurdistan Region/IRAQ.
3. Valerio Talamanca, Università Roma Tre
4. Michel Waldschmidt, Université Paris 6
From 12th-14th of March, The background of the main topic (Group, ring, polynomial ring and field) will be covered by the local lecturers
The main topics will be as follows
1.Topics in algebra, Pierre Cartier
2. Cyclotomy, Francesco Pappalardi
3.Mahler measure of polynomials, Valerio Talamanca
4.Introduction to Diophantine Approximation, Michel Waldschmidt
Description of each course
1. Cyclotomy
A Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum, a type of exponential sum which is a linear combination of periods.
– Reminders of the Galois Theory of cyclotomic polynomial
– Gauss sums and Gauss periods
– The determination of the quadratic Gauss Sum
– Gauss Theorem for expression of roots of unity via nested radicals
– The computation of
– Kummer’s problem on cubic Gauss periods
2. Mahler measure of polynomials
The Mahler measure of a polynomial (with integer coefficients) is a measure of its arithmetic complexity as well as the arithmetic complexity of its roots. Lehmer’s conjecture asserts that the Mahler measure of non cyclotomic polynomials is bounded below by an absolute constant. We willuse this very explicit problem to introduce a few concept of algebraic number theory.
– Mahler measure of a polynomial
– Algebraic integers and their minimal polynomial
– Absolute values and the height of algebraic numbers
– Lower bounds for the Mahler measure of totally real polynomials
3. Introduction to Diophantine Approximation
Approximation Rational approximation to a real number: continued fractions and applications. Rational approximation to an algebraic number: transcendence theorem of Thue,Siegel, Roth. Simultaneous approximation. Schmidt Subspace Theorem. Diophantine approximation in fields of power series.
