MA257: Introduction to Number Theory

✍ Scribed by J. E. Cremona

• Year: 2018
• Tongue: English
• Leaves: 48
• Series: lecture notes
• Edition: version 5 Jan 2018
• Category: Library

✦ Table of Contents

1. Introduction: What is Number Theory?......Page 2
   Basic Notation......Page 3
   1.1. Divisibility in Z......Page 4
   1.2. Greatest Common Divisors in Z......Page 5
   1.3. The Euclidean Algorithm in Z......Page 6
   1.4. Primes and unique factorization......Page 7
   1.5. Unique Factorization Domains......Page 9
   2.1. Definition and Basic Properties......Page 14
   2.2. The structure of Z/mZ......Page 15
   2.3. Euler's, Fermat's and Wilson's Theorems......Page 16
   2.4. Some Applications......Page 18
   2.5. The Chinese Remainder Theorem or CRT......Page 19
   2.6. The structure of Um......Page 21
   3.2. Legendre Symbols and Euler's Criterion......Page 23
   3.3. The Law of Quadratic Reciprocity......Page 24
   4.2. Sums of squares......Page 28
   4.3. Legendre's Equation......Page 30
   4.4. Pythagorean Triples......Page 32
   4.5. Fermat's Last Theorem......Page 33
   4.6. Proof of Minkowski's Theorem......Page 35
   5.1. Motivating examples......Page 37
   5.2. Definition of Zp......Page 38
   5.3. The ring Zp......Page 39
   5.4. The field Qp......Page 42
   5.5. Squares in Zp......Page 45
   5.6. Hensel lifting......Page 47