Overview
- Integrates abstract algebra into elementary number theory to enhance students' understanding
- Offer numerous IBL explorations that allow students develop insights for forthcoming material
- Written in an engaging, whimsical style to motivate and inspire students
Part of the book series: Undergraduate Texts in Mathematics (UTM)
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Table of contents (9 chapters)
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Front Matter
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Back Matter
Keywords
About this book
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.”
Authors and Affiliations
About the authors
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.
Bibliographic Information
Book Title: Explorations in Number Theory
Book Subtitle: Commuting through the Numberverse
Authors: Cam McLeman, Erin McNicholas, Colin Starr
Series Title: Undergraduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-3-030-98931-6
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2022
Hardcover ISBN: 978-3-030-98930-9Published: 19 December 2022
Softcover ISBN: 978-3-030-98933-0Published: 19 December 2023
eBook ISBN: 978-3-030-98931-6Published: 18 December 2022
Series ISSN: 0172-6056
Series E-ISSN: 2197-5604
Edition Number: 1
Number of Pages: XIII, 372
Number of Illustrations: 55 b/w illustrations, 6 illustrations in colour
Topics: Number Theory, General Algebraic Systems