An (algebraic) introduction to Number Theory (Fall 2017)

Description

Preface......Page 4 The dictionary answer......Page 6 Answered with questions......Page 7 Solutions and non-solutions......Page 10 Main branches of number theory......Page 14 Postscript: an example of elementary and analytic techniques......Page 16 References......Page 17 Standard number systems......Page 19 An axiomatic approach......Page 22 Beyond counting......Page 26 Rings and fields......Page 27 Binary operations......Page 28 Ring and field axioms......Page 31 Integers mod n......Page 36 Lapine numbers......Page 43 Quadratic rings......Page 46 Cyclotomic rings......Page 53 Beyond C......Page 57 Factorization......Page 60 Units, irreducibles and existence of factorizations......Page 61 Primes and unique factorization......Page 65 The Euclidean algorithm......Page 68 A Euclidean algorithm for (two) quadratic rings......Page 74 Unique factorization beyond Z......Page 79 Divisibility criteria......Page 85 Applications to Diophantine equations......Page 87 Groups and invertibility mod n......Page 91 Cosets and Lagrange's theorem......Page 96 RSA......Page 101 Sums of Two Squares......Page 107 Pythagorean Triples......Page 110 The Chinese Remainder Theorem......Page 112 Quadratic Reciprocity......Page 114 Numbers of the form x2+dy2......Page 120 Sums of three and four squares......Page 124 Units and Pell's equation......Page 131 Approximation and existence of solutions......Page 133 Fundamental units......Page 137 Continued fractions......Page 140 Aftermission: fundamental units and Fibonacci numbers......Page 145 The Last Theorem......Page 147 References......Page 150 Index......Page 151