Algebraic theory of quadratic numbers

206 Pages·2013·2.373 MB·English

by  Trifković, Mak

Universitext
Mak Trifković
Algebraic Theory
of Quadratic
Numbers
Universitext
Universitext
Series Editors:
SheldonAxler
San Francisco State University
VincenzoCapasso
Universit`a degli Studi di Milano
CarlesCasacuberta
Universitat de Barcelona
AngusJ.MacIntyre
Queen Mary, University of London
KennethRibet
University of California, Berkeley
ClaudeSabbah
CNRS, E´cole Polytechnique
EndreSu¨li
University of Oxford
WojborA.Woyczynski
Case Western Reserve University
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Mak Trifkovic´
Algebraic Theory of
Quadratic Numbers
123
MakTrifkovic´
DepartmentofMathandStatistics
UniversityofVictoria
Victoria,BC,Canada
ISSN0172-5939 ISSN2191-6675(electronic)
ISBN978-1-4614-7716-7 ISBN978-1-4614-7717-4(eBook)
DOI10.1007/978-1-4614-7717-4
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013941873
MathematicsSubjectClassification:11-01
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Preface
Elementary arithmetic studies divisibility and factorization of ordinary
integers.Algebraicnumber theoryconsidersthe samequestions for algebraic
numbers, solutions to polynomial equations with integer coefficients. In this
setting, (un)fortunately, the uniqueness of prime factorization no longer
holds. Remedying and measuring its failure is the starting point of algebraic
number theory.
There are many excellent texts both on elementary and on algebraic
number theory. However, I needed an intermediate-level book for an under-
graduatecoursewiththe aimofimpartingthe flavorandbeauty ofalgebraic
numbertheorywithminimalalgebraicprerequisites.Thenotesforthatcourse
grew into this book. Restricting to quadratic numbers was a natural choice:
hands-on examples abound, while the Galois theory is trivial to the point of
invisibility. Indeed, concreteness and computation underpin the approach of
the book. Computers are a great research tool; for learning, however, there
is no substitute for getting one’s hands dirty with calculations. A parallel
emphasisisontheinteractionbetweendifferentbranchesofmathematics,all
too often introducedto undergraduatesas separate worlds.In this book, the
student can see them joining forces to prove beautiful theorems. Linear and
abstract algebra together rescue unique factorization; basic plane geometry
is essential for proving that unique factorization never fails by much.
The prerequisites for this book are a knowledge of elementary number
theory and a passing familiarity with ring theory. Its goal is to give the
undergraduate a taste of algebraic number theory rightafter (or during) the
firstcourseinabstractalgebra,andbeforelearningGaloistheory.Indeed,the
bookcanserveasarichsourceofexamplesforaringtheorycourse.Abright
andbravefreshmancanreadit, withsomeeffort,evenbeforethe firstcourse
on rings: the necessary theory is covered, albeit briskly, in Chap.2. Finally,
the book can be a concrete supplement to a beginning graduate course in
algebraic number theory.
v
vi Preface
Here’s a brief outline of the contents.
Chapter 1 generalizes,to the extentpossible, the arithmetic ofZ to several
specificquadraticfields.Theexamplesarechosentopresenttherangeofnew
phenomena in algebraic number theory. The definitions of a ring and a field
are the only abstract algebra required here.
Chapter 2isaself-containedbutquickreviewofringtheory,withexamples
geared toward number theory.
Chapter 3 amplifies the standard theory of row- and column-reduction by
restricting to matrices with entries in Z.
Chapter 4 is the heart of the book. It develops algebraic number theory in
a general quadratic field, and culminates in the proofof unique factorization
of ideals.
Chapter 5 proves the finiteness of the ideal class group, with numerous
computational examples.
Chapter 6 completes the picture of the arithmetic of quadratic fields by
describingthegroupofunitsofarealquadraticfieldusingthetheoryofcon-
tinued fractions. This chapter and the next are arguably the most interest-
ing ones, since their techniques and results don’t generalize to higher-degree
fields.
Chapter 7 goes back to the roots of algebraic number theory—binary
quadratic forms. Forgoing the usual elementary presentation, we
emphasize the connections with ideals. The main result of the chapter is
a precise description of the identification of strict equivalence classes of
binary quadratic forms and narrow ideal classes. This chapter has no extra
prerequisites,but does requiremore mathematical maturity, as we introduce
the language of group actions and commutative diagrams. Concreteness is
not sacrificed, though: we cover in detail the algorithms for reducing both
definite and indefinite forms. The section concludes with a presentation of
Bhargava cubes, a recent development. The appendix assembles the results
on the orders in quadratic fields, proved in the exercises throughout the
book.
Thetermsbeingdefined,eitherinformaldefinitionsorintherunningtext,
are highlighted for ease of finding. All propositions, definitions, displayed
equations etc., are numbered in the same sequence. Starred exercises come
with a hint at the back of the book.
Thanks go to my department, which supported me in giving small ad-
vanced classes; to the students therein, who helped me test the approach of
the book;tomysummerstudents,ChrisWhitmanfortypingupthesenotes,
andJasperWiartforcarefullyreadingthemandcontributingtoChap.7;and
finally to Springer for their help and patience.
Victoria, BC, Canada Mak Trifkovi´c
Notation
General
|X| Size of finite set X
(cid:2)x(cid:3), (cid:4)x(cid:5) Nearest integer to x from below, resp. above
N,Z,Q,R,C Natural, integer, rational, real, complex
numbers
Z≥0 Non-negative integers, N∪{0}
a|b a divides b
X(cid:2) (cid:7)(cid:3)Y Disjoint union of sets X and Y
p Legendre symbol (Quadratic Reciprocity) p. 5
q
Algebra
ϕ:X −∼→Y ϕ is an isomorphism from X to Y
R/I Quotient ring p. 30
R[x] Ring of polynomials in x with coefficients in R p. 16
Gx Orbit of x under the action of G p. 145
X/G Quotient set for a group action of G on X p. 145
Quadratic Fields
√ √
Q[ D] Quadratic field F ={a+b D :a,b∈Q} p. 61
O Ring of integers in the field F p. 63
δ,t,n O =Z[δ], δ2−tδ+n=0 p. 65
Trα,Nα Trace and norm of α∈F p. 61
I+,I Set of nonzero ideals of O, resp. fractional p. 77
F F
ideals in F
Cl(F), Cl+(F) Ideal class group of F, resp. narrow class p. 87,152
group
h(F), h+(F) Class number of F, resp. narrow class number p. 87,152
vii
viii Notation
Continued Fractions
p /q ith convergent of a continued fraction p. 111
i i
η ith tail of the continued fraction of η p. 109
i
mi,vi Coefficients of ηi =(mi+ηO)/vi p. 118
Quadratic Forms
Q,Q ,Q Sets of quadratic forms p. 140,140
D F
H,H ,H Sets of quadratic numbers p. 140,140
D F
I Set of oriented fractional ideals in F p. 149
F (cid:4) (cid:5)
S,T 0−1 ,[11] as generators of SL (Z) p. 145
1 0 01 2
Contents
1 Examples................................................. 1
1.1 Review of Elementary Number Theory .................... 1
1.2 The Field Q[i] and the Gauss Integers..................... 8
1.3 Quadratic Int√egers...................................... 15
1.4 The Field Q[√−3] and the Eisenstein Integers.............. 17
1.5 The Field Q[√−5] ...................................... 20
1.6 The Field Q[ 319] ..................................... 23
2 A Crash Course in Ring Theory .......................... 27
2.1 Basic Definitions ....................................... 27
2.2 Ideals, Homomorphisms, and Quotients ................... 29
2.3 Principal Ideals ........................................ 33
2.4 Operations on Ideals.................................... 36
2.5 Prime and Maximal Ideals............................... 41
3 Lattices................................................... 45
3.1 Group Structure of Lattices.............................. 45
3.2 Linear Algebra Over Z .................................. 48
3.3 Computing with Ideals .................................. 52
3.4 Lattice Quotients....................................... 55
√
4 Arithmetic in Q[ D] ..................................... 61
4.1 Quadratic Fields ....................................... 61
4.2 The Ring of Integers .................................... 63
4.3 Unique Factorization of Ideals: The Road Map ............. 67
4.4 Noether Rings ......................................... 68
4.5 Standard Form of an Ideal............................... 70
4.6 The Ideal Norm ........................................ 73
4.7 Fractional Ideals ....................................... 77
4.8 Unique Factorization of Ideals............................ 80
4.9 Prime Ideals in O ...................................... 81
ix