Explorations in Number Theory: Commuting through the Numberverse

This undergraduate textbook approaches number theory through the lens of abstract algebra,using inquiry-based learning (IBL) and Python worksheets to introduce students to rings, groups, fields, and other algebraic structures. It is organized around three core themes and includes exercises and IBL explorations in public key cryptography, Lagranges four-square theorem, units and Pells Equation, Fermats Last Theorem, and other deeper mysteries of algebraic number theory.

Format: Hardback
Length: 372 pages
Publication date: 19 December 2022
Publisher: Springer Nature Switzerland AG

This groundbreaking undergraduate textbook delves into the realm of number theory by employing the framework of abstract algebra. Written in an engaging and whimsical style, this text seamlessly introduces students to rings, groups, fields, and various algebraic structures, while unraveling the fundamental concepts of elementary number theory. Throughout the chapters, inquiry-based learning (IBL) takes center stage, fostering a deep understanding of the subject matter while simultaneously strengthening students' grasp of previously covered topics.

The text is structured around three central themes:

The concept of what a "number" truly entails and the realization that a comprehensive exploration of number theory necessitates a thorough familiarity with a diverse array of number systems.

The utilization of Diophantine equations as powerful catalysts for introducing and developing structural ideas within the realm of number theory.

The profound role of abstract algebra in number theory, particularly in elucidating the Fundamental Theorem of Arithmetic for various novel number systems.

In addition to these core themes, the textbook also encompasses various other aspects of modern number theory. These include the study of elliptic curves, the analogies between integer and polynomial arithmetic, p-adic arithmetic, and the intricate relationships between the spectra of primes in different rings. These threads are skillfully woven throughout the chapters and exercise sets, adding depth and richness to the learning experience.

To enhance student engagement, each chapter concludes with a comprehensive array of exercises. These exercises are categorized into four distinct sections: Calculations and Informal Proofs, Formal Proofs, Computation and Experimentation, and General Number Theory Awareness. IBL "Exploration" worksheets are strategically placed in numerous sections, some of which involve numerical investigations. Furthermore, to assist students who may not have prior experience with programming languages, Python worksheets are readily accessible on the book's website.

In the final chapter, a series of five additional IBL explorations are presented, offering further reinforcement and expansion of the concepts learned throughout the text. These explorations delve into diverse topics, encouraging students to apply their knowledge and explore number theory in new and exciting ways.

In summary, this innovative undergraduate textbook offers a unique and captivating approach to number theory, blending abstract algebra with practical applications. Through its engaging writing style, IBL methodology, and comprehensive exercise sets, it equips students with the necessary tools to excel in the field and unlock the mysteries of number theory. Whether you are a student of mathematics or simply a curious mind seeking to expand your knowledge, this textbook is an invaluable resource for your academic journey.

Weight: 762g
Dimension: 161 x 241 x 27 (mm)
ISBN-13: 9783030989309
Edition number: 1st ed. 2022