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

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