Theory of Numbers Divisibility Theory in the Integers, The Theory of Congruences, Number-Theoretic Functions, Primitive Roots, Quadratic Residues Yotsanan Meemark Informal style based on the course 2301331 Theory of Numbers, offered at Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University Second version August 2016 Any comment or suggestion, please write to
[email protected] Contents 1 Divisibility Theory in the Integers 1 1.1 The Division Algorithm and GCD . 1 1.2 The Fundamental Theorem of Arithmetic . 5 1.3 The Euclidean Algorithm and Linear Diophantine Equations . 8 2 The Theory of Congruences 13 2.1 Basic Properties of Congruence . 13 2.2 Linear Congruences . 16 2.3 Reduced Residue Systems . 18 2.4 Polynomial Congruences . 21 3 Number-Theoretic Functions 25 3.1 Multiplicative Functions . 25 3.2 The Mobius¨ Inversion Formula . 28 3.3 The Greatest Integer Function . 30 4 Primitive Roots 33 4.1 The Order of an Integer Modulo n ............................ 33 4.2 Integers Having Primitive Roots . 35 4.3 nth power residues . 39 4.4 Hensel’s Lemma . 41 5 Quadratic Residues 45 5.1 The Legendre Symbol . 45 5.2 Quadratic Reciprocity . 48 Bibliography 53 Index 54 Chapter 1 Divisibility Theory in the Integers Let N denote the set of positive integers and let Z be the set of integers. 1.1 The Division Algorithm and GCD Theorem 1.1.1. [Well-Ordering Principle] Every nonempty set S of nonnegative integers contains a least element; that is, there is some integer a in S such that a ≤ b for all b 2 S.