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E-raamat: Explorations in Number Theory: Commuting through the Numberverse

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Explorations in Number Theory: Commuting through the NumberverseSuurem pilt
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This innovative undergraduate textbook approaches number theory through the lens of abstract algebra.  Written in an engaging and whimsical style, this text will introduce students to rings, groups, fields, and other algebraic structures as they discover the key concepts of elementary number theory.  Inquiry-based learning (IBL) appears throughout the chapters, allowing students to develop insights for upcoming sections while simultaneously strengthening their understanding of previously covered topics.
  
The text is organized around three core themes: the notion of what a “number” is, and the premise that it takes familiarity with a large variety of number systems to fully explore number theory; the use of Diophantine equations as catalysts for introducing and developing structural ideas; and the role of abstract algebra in number theory, in particular the extent to which it provides the Fundamental Theorem of Arithmetic for various new number systems.  Other aspects of modern number theory – including the study of elliptic curves, the analogs between integer and polynomial arithmetic, p-adic arithmetic, and relationships between the spectra of primes in various rings – are included in smaller but persistent threads woven through chapters and exercise sets.

Each chapter concludes with exercises organized in four categories: Calculations and Informal Proofs, Formal Proofs, Computation and Experimentation, and General Number Theory Awareness.  IBL “Exploration” worksheets appear in many sections, some of which involve numerical investigations.  To assist students who may not have experience with programming languages, Python worksheets are available on the book’s website.  The final chapter provides five additional IBL explorations that reinforce and expand what students have learned, and can be used as starting points for independent projects.  The topics covered in these explorations are public key cryptography, Lagrange’s four-square theorem, units and Pell’s Equation, various cases of the solution to Fermat’s Last Theorem, and a peek into other deeper mysteries of algebraic number theory.

Students should have a basic familiarity with complex numbers, matrix algebra, vector spaces, and proof techniques, as well as a spirit of adventure to explore the “numberverse.”
1 What is a Number?
1(18)
1.1 Human conception of numbers
1(3)
1.2 Algebraic Number Systems
4(1)
1.3 New Numbers, New Worlds
5(8)
1.4 Exercises
13(6)
References
18(1)
2 A Quick Survey of the Last Two Millennia
19(18)
2.1 Fermat, Wiles, and The Father of Algebra
19(2)
2.2 Quadratic Equations
21(6)
2.3 Diophantine Equations
27(6)
2.4 Exercises
33(4)
References
35(2)
3 Number Theory in Z Beginning
37(42)
3.1 Algebraic Structures
37(7)
3.2 Linear Diophantine Equations and the Euclidean Algorithm
44(13)
3.3 The Fundamental Theorem of Arithmetic
57(6)
3.4 Factors and Factorials
63(7)
3.5 The Prime Archipelago
70(3)
3.6 Exercises
73(6)
4 Number Theory in the Mod-n Era
79(58)
4.1 Equivalence Relations and the Binary World
79(5)
4.2 The Ring of Integers Modulo n
84(4)
4.3 Reduce First and ask Questions Later
88(5)
4.4 Division, Exponentiation, and Factorials in Z/(n)
93(7)
4.5 Group Theory and the Ring of Integers Modulo n
100(8)
4.6 Lagrange's Theorem and the Euler Totient Function
108(10)
4.7 Sunzi's Remainder Theorem and φ(n)
118(7)
4.8 Phis, Polynomials, and Primitive Roots
125(3)
4.9 Exercises
128(9)
5 Gaussian Number Theory: Z[ i] of the Storm
137(34)
5.1 The Calm Before
137(1)
5.2 Gaussian Divisibility
138(6)
5.3 Gaussian Modular Arithmetic
144(3)
5.4 Gaussian Division Algorithm: The Geometry of Numbers
147(4)
5.5 A Gausso-Euclidean Algorithm
151(4)
5.6 Gaussian Primes and Prime Factorizations
155(5)
5.7 Applications to Diophantine Equations
160(5)
5.8 Exercises
165(6)
6 Number Theory, from Where We R to Across the C
171(38)
6.1 From --- 1 to ---d
171(3)
6.2 Algebraic Numbers and Rings of Integers
174(6)
6.3 Quadratic Fields: Integers, Norms, and Units
180(6)
6.4 Euclidean Domains
186(6)
6.5 Unique Factorization Domains
192(3)
6.6 Euclidean Rings of Integers
195(10)
6.7 Exercises
205(4)
7 Cyclotomic Number Theory: Roots and Reciprocity
209(36)
7.1 Introduction
209(2)
7.2 Quadratic Residues and Legendre Symbols
211(3)
7.3 Quadratic Residues and Non-Residues Mod p
214(2)
7.4 Application: Counting Points on Curves
216(4)
7.5 The Quadratic Reciprocity Law: Statement and Use
220(3)
7.6 Some Unexpected Helpers: Roots of Unity
223(5)
7.7 A Proof of Quadratic Reciprocity
228(9)
7.8 Quadratic UFDs
237(3)
7.9 Exercises
240(5)
8 Number Theory Unleashed: Release Zp!
245(54)
8.1 The Analogy between Numbers and Polynomials
245(4)
8.2 The p-adic World: An Analogy Extended
249(6)
8.3 p-adic Arithmetic: Making a Ring
255(7)
8.4 Which numbers are p-adic?
262(4)
8.5 Hensel's Lemma
266(7)
8.6 The Local-Global Philosophy and the Infinite Prime
273(4)
8.7 The Local-Global Principle for Quadratic Equations
277(8)
8.8 Computations: Quadratic Equations Made Easy
285(5)
8.9 Synthesis and Beyond: Moving Between Worlds
290(3)
8.10 Exercises
293(6)
9 The Adventure Continues
299(56)
9.1 Exploration: Fermat's Last Theorem for Small n
301(7)
9.2 Exploration: Lagrange's Four-Square Theorem
308(10)
9.3 Exploration: Public Key Cryptography
318(17)
9.3.1 Public Key Encryption: RSA
320(5)
9.3.2 Elliptic Curve Cryptography
325(8)
9.3.3 Elliptic ElGamal Public Key Cryptosystem
333(2)
9.4 Exploration: Units of Real Quadratic Fields
335(7)
9.5 Exploration: Ideals and Ideal Numbers
342(11)
9.6 Conclusion: The Numberverse, Redux
353(2)
Appendix I Number Systems 355(10)
Index 365(6)
Index of Notation 371
Cam McLeman is Associate Professor of Mathematics at the University of Michigan-Flint. Erin McNicholas is Professor of Mathematics at Willamette University in Salem, Oregon. Colin Starr is a Professor of Mathematics at Willamette University in Salem, Oregon.
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