LEADER 04648cam 2200493 i 4500001 on1301899746 003 OCoLC 005 20230214081902.1 008 220304s2022 sz a b 001 0 eng d 019 13014850061301766933 020 9783030989309 |qhardcover 020 3030989305 |qhardcover 020 |z9783030989316 |qelectronic book 035 (Sirsi) 40031554856 035 40031554856 035 (OCoLC)1301899746 |z(OCoLC)1301485006 |z(OCoLC)1301766933 040 YDX |beng |erda |cYDX |dBDX |dUKMGB |dOCLCF |dOHX |dYDX |dUtOrBLW 050 4 QA241 |b.M43 2022 082 04 512.74 |223 100 1 McLeman, Cam, |eauthor. |?UNAUTHORIZED 245 10 Explorations in number theory : |bcommuting through the numberverse / |cCam McLeman, Erin McNicholas, Colin Starr. 264 1 Cham, Switzerland : |bSpringer, |c[2022] 300 xiii, 372 pages : |billustrations (some color) ; |c24 cm. 336 text |2rdacontent 336 still image |2rdacontent 337 unmediated |2rdamedia 338 volume |2rdacarrier 490 1 Undergraduate texts in mathematics 505 0 Preface -- What is a Number?- A Quick Survey of the Last Two Millenia -- Number Theory in $\mathcal{Z}$ Beginning -- Number Theory in the Mod-n Era -- Gaussian Number Theory: $\mathcal{Z}[i]$ of the Storm -- Number Theory: From Where We $\mathcal{R}$ to across the $mathcal{C}$ -- Cyclotomic Number Theory: Roots and Reciprocity. Number Theory Unleashed: Release $\mathcal{Z}_p$!- The Adventure Continues -- Appendix: Number Systems. 520 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". 504 Includes bibliographical references and indexes. 650 0 Number theory. |=^A22877 650 0 Algebraic number theory. |=^A23234 650 0 Algebra, Abstract. |=^A42913 650 7 Algebra, Abstract. |2fast |0(OCoLC)fst00804919 650 7 Algebraic number theory. |2fast |0(OCoLC)fst00804937 700 1 McNicholas, Erin, |eauthor. |?UNAUTHORIZED 700 1 Starr, Colin |q(Colin L.), |eauthor. |=^A1454210 830 0 Undergraduate texts in mathematics. |=^A236098 949 |i30372017358612 |ojjlm 960 |o1 |s69.99 |tJoyner48 |uJAPP |zUSD 596 1 998 5972499