Explorations in Number Theory: Commuting through the Numberverse

380 Pages·2022·3.639 MB·English

by  Cam McLeman, Erin McNicholas, Colin Starr

Undergraduate Texts in Mathematics
Cam McLeman
Erin McNicholas
Colin Starr
Explorations
in Number
Theory
Commuting through the Numberverse
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
Series Editors
Pamela Gorkin
Mathematics Department, Bucknell University, Lewisburg, PA, USA
Jessica Sidman
Mathematics and Statistics, Mount Holyoke College, South Hadley, MA, USA
Advisory Board
Colin Adams, Williams College, Williamstown, MA, USA
Jayadev S. Athreya, University of Washington, Seattle, WA, USA
Nathan Kaplan, University of California, Irvine, CA, USA
Lisette G. de Pillis, Harvey Mudd College, Claremont, CA, USA
Jill Pipher, Brown University, Providence, RI, USA
Jeremy Tyson, University of Illinois at Urbana-Champaign, Urbana, IL, USA
Undergraduate Texts in Mathematics are generally aimed at third- and
fourth-year undergraduate mathematics students at North American universities.
Thesetextsstrivetoprovidestudentsandteacherswithnewperspectivesandnovel
approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation
ofinterrelations among different aspectsofthesubject.Theyfeature examples that
illustrate key concepts as well as exercises that strengthen understanding.
Cam McLeman (cid:129) Erin McNicholas (cid:129)
Colin Starr
Explorations in Number
Theory
Commuting through the Numberverse
123
CamMcLeman ErinMcNicholas
Department ofMathematics Department ofMathematics
University of Michigan–Flint Willamette University
Flint, MI,USA Salem, OR,USA
Colin Starr
Department ofMathematics
Willamette University
Salem, OR,USA
ISSN 0172-6056 ISSN 2197-5604 (electronic)
Undergraduate Texts inMathematics
ISBN978-3-030-98930-9 ISBN978-3-030-98931-6 (eBook)
https://doi.org/10.1007/978-3-030-98931-6
MathematicsSubjectClassification: 11-01
©SpringerNatureSwitzerlandAG2022
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Preface
...wherein we welcome readers of all sorts to our journey.
Who is This Text’s Audience?
We have written this book for you, our inquisitive, delightful reader! Your dis-
cerningeyeformathematics,interestinnumbers,willingnesstoexplore,anddesire
to understand underlying structures deeply are commendable. While this text
examinesnumbersystemsfromanalgebraiclens,nopreviousexposuretoabstract
algebra is assumed. We do, however, assume some familiarity with vector spaces
andbasicprooftechniques.Calculus:MultivariablebyMcCallumetal.,andProofs
and Fundamentals: A First Course in Abstract Mathematics by Ethan Bloch are
excellent references for the assumed background material.
To the Student:
Dear Student,
Welcome to the start of your journey! Consider this book as your official
companionguide—itisinsomewaysatraditionalmathematicstextbook(ithas,for
example, lots of math in it), but deviates from standard practice in several notable
ways as well. First, it has multiple authors coming from a variety of backgrounds
andexperienceswithinthegeneralrealmofalgebraandnumbertheory.Youmight
see sparks of more excited writing as one of us encountered a favorite piece of
numbertheory,forexample,thoughwegenerallyattemptedtoconformourwriting
to a single voice1.
Second, and more importantly, we call particular attention to the existence of
“Explorations” sprinkled through the chapters. These worksheets are the manifes-
tations of the authors’ zeal for inquiry-based learning—the process offiguring out
mathematics for yourself like our mathematical forebears out in the unexplored
1Well,someofusdid.
v
vi Preface
wilderness. Thereareno solutionsprovided totheworksheets, astheirpoint isnot
typicallyto“gettheanswer” butrather tobring up questions that we’ll spend time
discussinginthefollowingsection.Sodon’tskipthem!Infact,youmightwantto
photocopy those pages for repeated use. Or better yet, buy several copies of the
book, just to be safe.
To the Instructor:
Dear Instructor,
Phew! Now that the students have stopped reading, we can be frank. Two
adjectives that stand out as highly desirable when choosing a textbook are com-
prehensiveandself-contained.Thisbookdoesnotattempttoachieveeitherofthose
ambitions. In exchange, we stake a claim to the partially opposite adjectives of
being modern and flexible. As to comprehensiveness, the over-arching goal of the
texthasbeentohookthereader’sinterestsinthemysteryofnumbers,anditisthe
opinion of the authors that no book this size can do justice to the vast wealth of
classical number-theoretic pearls while simultaneously working toward contem-
porarynumbertheory.Consequently,our“toincludeornottoinclude”philosophy
hasbeentofocusondevelopingafewcorethemesthatpersistthroughthestudyof
number theory at all levels:
(cid:129) The study of Diophantine equations, partly for their own sake, but more typi-
cally as catalysts for introducing and developing bigger structural ideas.
(cid:129) The notion of what a “number” is, and the premise that it takes familiarity in
quite a large variety of number systems to fully explore number theory.
(cid:129) The use of abstract algebra in number theory, and in particular the extent to
which it provides the “Fundamental Theorem of Arithmetic” for various new
number systems.
Inadditiontothecorethemes,otheraspectsofmodernnumbertheoryarepresentin
smaller but persistent threads woven through chapters and exercise sets, e.g., the
studyofellipticcurves,theanalogiesbetweenintegerandpolynomialarithmetic,p-
adic valuations,relationshipsbetweenthespectrum ofprimes invariousrings,etc.
Thepropersequencingofnumbertheoryandabstractalgebraisaconundrumwe
have chosen to embrace. While an introduction to number theory provides a good
preparationforthestudyofabstractalgebra,studentswhohavehadabstractalgebra
arebetterabletofullygrasptheconceptsinanintroductorynumbertheorycourse.
Our solution is to develop and define algebraic concepts as needed to summarize
and formalize the patterns observed in various number systems. In this text, stu-
dents encounter groups, rings, fields, ideals, and more. While we have taken
uncharacteristic care to make the coverage of these concepts self-contained, it is
certainly not comprehensive. In our experience, the result is a text that simulta-
neously serves both students with and without any previous exposure to abstract
algebra. Students with little to no abstract algebra experience are able to grasp the
Preface vii
salient details by applying the intuition and experience gained from the adjacent
number theory. These students gain a solid foundation on which to build in a
subsequentabstractalgebracourse.Forstudentswhohavehadacourseinabstract
algebra, the pithy and targeted coverage of algebraic concepts is a good refresher
that helps them apply their algebraic knowledge to gain a deeper understanding
of the number theory content. For these students, the text provides not only an
introductiontonumbertheorybutalsoastrengthenedappreciationforthepowerof
an algebraic perspective.
As to self-containedness more broadly, we have been content to assume some
reasonable prerequisites and avoid compulsively writing appendices to provide
basic familiarity with complex numbers, matrix algebra, and vector spaces. In
additiontodirectingstudentstogoodtextsourcesforthisbackgroundmaterial,we
take advantage of a thoroughly modern approach, namely, the ability of readers to
go online and look things up. The skill of finding and processing mathematics
onlineisanincreasinglycrucialoneandshouldbefosteredratherthandiscouraged.
This philosophy makes itself explicit in the form of a section of each chapter’s
homeworkproblems(“GeneralNumberTheoryAwareness”problems)whosegoal
is to get students to conduct research online on biographical information about
mathematicians and mathematical content that space prevented us from including,
or pieces of mathematical culture that don’t fit strictly in the traditional academic
viewofwhatamathematicstextbookshouldcover.Exercisesinthesesectionsalso
leadstudentstopracticeoneofthemostfundamentalmathematicalresearchskills:
seeing,uponmakingamathematicaldiscovery,ifsomeoneelsehasalreadydoneit.
The broader claim to modernity is an over-arching belief in inquiry-based
learning. In addition to these research-based homework questions, a second sub-
sectionofexerciseshasstudentsnumericallyinvestigateconjecturesandmaketheir
own. Sometimes, the tools to prove these conjectures will be within a student’s
grasp, and sometimes they will be cutting-edge research questions to which
humanity does not know the answer. While we do not prescribe a programming
language for these numerical investigations, the book’s companion website has
Python worksheets designed to walk students with little to no programming
experience through those exercises. Finally, and perhaps most importantly,
inquiry-based learning is built into the text itself: exploration worksheets lovingly
placed throughout the chapters let students lead the way by coming up with the
pivotal ideas for the upcoming sections while simultaneously improving their flu-
encyinpreviouslycoveredtopics.Thefinalchaptercloselymirrorsthatofapurely
inquiry-based textbook, allowing students to work individually or in groups—
perhaps as a final project—through problem-driven sections covering material
begun in the main text (Fermat’s Last Theorem, quaternions, real quadratic units,
elliptic curves, cryptography, ideal theory). Exercises designed to pique students’
interest in these topics are interspersed throughout earlier chapters.
Thepacingofthecourseisalsoratherflexible.Thedefaultapproach,webelieve,
should begin with the standard theoretical development of Z and Z=ðnÞ but using
the explicit language of groups and rings, rather than shunting this perspective off
asadvancedmaterialneartheendofthecourse.Incorporatingabstractalgebrainto
viii Preface
thisprocessnecessarilyproceedsmoreslowly,buthasthebenefitofbeingquickly
generalizable: once you prove that Z½i(cid:2) has a Euclidean algorithm, for example,
proving that it enjoys unique factorization is essentially a matter of copy–pasting
theargumentfromZ.GoingquicklythroughChapters5and6leavesenoughtime
to do both culminating chapters, on quadratic reciprocity and p-adic numbers, in a
singlesemester.Ontheotherhand,aninstructorwhowantedtousethebooktofill
twosemesterscoveringbothabstractalgebraandnumbertheorycouldeasilydoso
by using all of the in-class explorations, covering the details of abstract Euclidean
domains, and providing in-class time for student presentations from Chapter 9.
Finally, some closing remarks on the use of the book. As mentioned in the
student preface, we highly value the inclusion of the explorations appearing
throughout the book. They provide an inquiry-based break from the more tradi-
tional class structure during which an instructor can engage in informal formative
assessmentofhowtheclassisdoingandallowstudentstoengageinindependentor
group discovery. Questions andanswers tothese problemsaretypicallydeveloped
more carefully in the sections that follow. Relatedly, the homework addresses a
variety of ways in which one becomes a master of number theory. There are
computational exercises to promote fluency in arithmetic calculations, proof-based
problems to develop theoretical understanding, theaforementioned online research
problems, and problems that lend themselves to computer-aided exploration.
Suggested Pacing and Content Coverage
A note on the use of explorations: we find that different classes take significantly
different amounts of time to work through the items on each exploration.
Accordingly, we do not necessarily recommend trying to get through a full
explorationinclass. Partscould beomittedorassignedasadditional homeworkas
needed.
For a 14-week semester, 3 hours per week, we suggest the following pacing:
(cid:129) Week 1: Sections 1.1–2.3 (leaving most details for students to read),
Exploration A.
(cid:129) Week 2: Section 3.1, Exploration B, and Section 3.2.
(cid:129) Week 3: Exploration C, Sections 3.3 and 3.4, Exploration D, and Section 3.5.
(cid:129) Week 4: Sections 4.1–4.3, Exploration E, and Section 4.4.
(cid:129) Week 5: Exploration F, Section 4.5, Exploration G, and Section 4.6.
(cid:129) Week 6: Exploration H, Sections 4.7 and 4.8.
(cid:129) Week 7: Sections 5.1 and 5.2, Exploration I, Sections 5.3 and 5.4, and
Exploration J.
(cid:129) Week 8: Section 5.5, Exam 1, and Section 5.6.
(cid:129) Week 9: Sections 5.6–6.2, Exploration K, and Section 6.3.
(cid:129) Week 10: Sections 6.3 and 6.4, Exploration L, and Sections 6.5 and 6.6.
Preface ix
(cid:129) Week 11: Sections 7.1 and 7.2, Exploration M, Sections 7.3 and 7.5.
(cid:129) Week 12: Sections 7.5–7.7.
(cid:129) Week 13: Chapter 9 topics.
(cid:129) Week 14: Chapter 9 topics.
Flint, USA Cam McLeman
Salem, USA Erin McNicholas
Salem, USA Colin Starr
AcknowledgmentsWeappreciatetheeffortsofoureditorstokeepusontrackandthehelpfuland
insightfulcommentsofthereviewers.Additionally,eachoftheauthorswouldliketoexpresstheir
heartfeltappreciationtotheothertwoauthorsforeachdoingapproximately25%ofthework.