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Explorations in Number Theory - Commuting through the Numberverse (Hardcover, 1st ed. 2022)
Loot Price: R1,496
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Explorations in Number Theory - Commuting through the Numberverse (Hardcover, 1st ed. 2022)
Series: Undergraduate Texts in Mathematics
Expected to ship within 9 - 15 working days
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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."
General
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