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MCLEMAN:EXPLORATIONS IN NUMBER THEORY

McNicholas, Erin
SPRINGER
12 / 2023
9783030989330
Inglés
Undergraduate Texts in Mathematics
ESPAÑA
Algebra
Teoria de numeros
Ingles

Sinopsis

This innovative undergraduate textbook approaches number theory through the lens of abstract algebra.nbsp; 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.nbsp; 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.nbsp;nbsp;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.nbsp; 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.nbsp; IBL ?Exploration? worksheets appear in many sections, some of which involve numerical investigations.nbsp; To assist students who may not have experience with programming languages, Python worksheets are available on the book's website.nbsp; 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.nbsp; 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.?

PVP
62,39€