A Journey Through The Realm of Numbers: From Quadratic Equations to Quadratic Reciprocity

Preface
Leitfaden
Contents
Outlooks
Notation
Chapter 1 Introduction: Polynomial Equations
	1.1 Shining a Light on the Midnight Formula
	1.2 The Cubic Equation
		1.2.1 Some History
		1.2.2 Solving the Cubic Equation
		1.2.3 Some Examples
		1.2.4 A Short Reflection
	1.3 From the Natural to the Real Numbers
		1.3.1 The Natural Numbers
		1.3.2 The Integers
		1.3.3 The Rational Numbers
		1.3.4 An Irrational Number
		1.3.5 The Real Numbers
		1.3.6 A Cubic Equation Always has a Real Solution
		1.3.7 Pythagoras\' Theorem
		1.3.8 Proofs and Training
	1.4 Complex Numbers
		1.4.1 Defining Complex Numbers
		1.4.2 The Geometry of Complex Numbers
		1.4.3 Doing Algebra in Polar Coordinates
		1.4.4 Becoming Multi-Lingual about Complex Numbers
		1.4.5 What is Lost?
	1.5 Returning to Polynomial Equations
		1.5.1 The Quadratic Equation
		1.5.2 The Cubic Equation
		1.5.3 The Quartic Equation: Ferrari\'s Method
	1.6 Is There No End?
		1.6.1 Gauss\' Theorem
		1.6.2 Algebraic Numbers
	1.7 Problems
	1.8 Programming
		1.8.1 SageMath as a Calculator
		1.8.2 First Scripts
		1.8.3 Lists and Loops
		1.8.4 Recursion
		1.8.5 Plotting
		1.8.6 Projects
Chapter 2 Cantor\'s Paradise
	2.1 Sets
	2.2 Cardinality of Sets
		2.2.1 Finite Cardinalities
		2.2.2 Cardinalities of Sets
		2.2.3 Functions, Injections, and Bijections
		2.2.4 Seating the Concert Guests
		2.2.5 Counting Infinite Sets
	2.3 Hilbert\'s Hotel
	2.4 Cantor\'s Diagonal Argument
	2.5 The Schröder–Bernstein Theorem
	2.6 Is There No End?
	2.7 Ordinal Numbers
	2.8 Problems
	2.9 Programming
		2.9.1 Tuples, Lists, and Sets
		2.9.2 Projects
Chapter 3 Sums of Squares
	3.1 Studying Sums of Squares
		3.1.1 Two Related Questions
	3.2 Modular Arithmetic
		3.2.1 Remainders and Divisibility
		3.2.2 Congruences
	3.3 Mathematical Induction
		3.3.1 Strong Induction
		3.3.2 Prime Factorization in N
	3.4 Sums of Squares: First Thoughts
		3.4.1 Four Squares
		3.4.2 Three Squares
		3.4.3 Two Squares
		3.4.4 Difference of Two Squares
	3.5 Sums of Squares: First Answers
		3.5.1 Pythagorean Triples
		3.5.2 Four Squares Again
		3.5.3 Three Squares Again
		3.5.4 Two Squares Again
		3.5.5 A Short Reflection
	3.6 Problems
	3.7 Programming
Chapter 4 Sums of Two Squares
	4.1 Divisibility and Primes in Gaussian Integers
		4.1.1 Primality and Irreducibility
		4.1.2 Division with Remainder
		4.1.3 Greatest Common Divisor
		4.1.4 Modular Arithmetic
		4.1.5 Irreducible Elements are Prime
	4.2 Writing Primes as Sums of two Squares
		4.2.1 Analyzing Fp
		4.2.2 Primes Congruent to 1 Modulo 4
	4.3 Zagier\'s One-Sentence Proof
		4.3.1 The Windmill Interpretation for Triples in S
		4.3.2 The Windmill Operation
		4.3.3 The Windmill Operation is an Involution
		4.3.4 Returning to the set S of Integer Triples
	4.4 Primality and Irreducibility
		4.4.1 Consequences of Unique Factorization
	4.5 Primes in the Gaussian Integers
		4.5.1 Which Primes Remain Prime?
		4.5.2 Completing the Description of Gaussian Primes
	4.6 The Proof for Sums of Two Squares
	4.7 More Examples
		4.7.1 +2
		4.7.2 +3
		4.7.3 +5
	4.8 Problems
	4.9 Programming
Chapter 5 Abstract Algebra: Ring Theory
	5.1 Abstraction
		5.1.1 Commutative Rings
		5.1.2 Euclidean Domains
		5.1.3 Fields
		5.1.4 A Remark on Proofs and the Journey Ahead
	5.2 The Polynomial Ring
		5.2.1 Primes in Polynomial Rings
	5.3  Euclidean Domains
		5.3.1 Modular Arithmetic
		5.3.2 The Residue Field
		5.3.3 The Chinese Remainder Theorem
	5.4 Problems
	5.5 Programming
Chapter 6 Cubic and Quartic Diophantine Equations
	6.1 Another Look at Euclid\'s Formula
	6.2 The Quartic Case of Fermat\'s Last Theorem
	6.3 A Fixed Gap Between a Square and a Cube
	6.4 The Cubic Case of Fermat\'s Last Theorem
	6.5 A Challenge for Eisenstein Integers and its Consequences
		6.5.1 Cubes One Below Squares
		6.5.2 A Twisted Cubic Fermat Theorem
	6.6 Problems
		6.6.1 The Cubic Equation Challenge
	6.7 Programming
Chapter 7 The Structure of the Group Fp
	7.1 (Abelian) Groups
		7.1.1 Abelian Groups
		7.1.2 Order of Elements
		7.1.3 Multiplicative Groups of Fields
		7.1.4 Cyclic Groups
		7.1.5 Non-abelian Groups
		7.1.6 Lagrange\'s Theorem
	7.2 The Multiplicative Group Fp
		7.2.1 Squares in Fp
		7.2.2 Proving Cyclicity
		7.2.3 Is There No End?
	7.3 Number Theory Public Key Exchange Algorithms
		7.3.1 One-Way Functions
		7.3.2 Diffie–Hellman Public Key Exchange
	7.4 Problems
	7.5 Programming
Chapter 8 Studying Squares Again
	8.1 Quadratic Reciprocity
		8.1.1 Proof of Quadratic Reciprocity
		8.1.2 A Summary Using the Quadratic Symbol
	8.2 More Examples
		8.2.1 The Euclidean Domain R
		8.2.2 Primes in R
		8.2.3 Factorizing +d
		8.2.4 A Square Modulo p
		8.2.5 Completing the Proof using Quadratic Reciprocity
	8.3 An Example with Switched Sign
		8.3.1 The Euclidean Domain Z[2]
		8.3.2 Primes in Z[2]
		8.3.3 Factorizing -2
		8.3.4 A Square Modulo p
		8.3.5 Completing the Proof using Quadratic Reciprocity
	8.4 The Challenge +5
	8.5 Problems
		8.5.1 Characterizing -3
		8.5.2 Characterizing -5
		8.5.3 The Challenge +5
		8.5.4 The Challenge +7
		8.5.5 The Challenge +11
		8.5.6 The Challenge x2=y3-19
	8.6 Programming
Hints to Selected Exercises
References and Further Reading
Index