Advanced OpenAccess ORM Questions and Answers
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Advanced OpenAccess ORM Questions and Answers

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Telerik TV | Video Transcript Page 1 of 15 ADVANCED OPENACCESS ORM QUESTIONS AND ANSWERS Video Transcript – July 22, 2009 Todd Anglin: Welcome everyone to a special format webinar here at Telerik. It is actually a recorded interview and we got a couple of OpenAccess and ORM Data experts here joining me today. My name is Todd Anglin, I am the Chief Evangelist at Telerik, but joining me on the line is Telerik Chief Strategy Officer, Stephen Forte, hello Stephen … Stephen Forte: Hi there Todd.
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Algebraic Number Theory,
a Computational Approach
William Stein
September 26, 20112Contents
1 Introduction 9
1.1 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 What is algebraic number theory? . . . . . . . . . . . . . . . . . . . 9
1.2.1 Topics in this book . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Some applications of algebraic number theory . . . . . . . . . . . . . 10
I Algebraic Number Fields 13
2 Basic Commutative Algebra 15
2.1 Finitely Generated Abelian Groups . . . . . . . . . . . . . . . . . . . 15
2.2 Noetherian Rings and Modules . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 The Ring Z is noetherian . . . . . . . . . . . . . . . . . . . . 23
2.3 Rings of Algebraic Integers . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Norms and Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Recognizing Algebraic Numbers using Lattice Basis Reduction (LLL) 33
2.5.1 LLL Reduced Basis . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.2 What LLL really means . . . . . . . . . . . . . . . . . . . . . 35
2.5.3 Applying LLL . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Dedekind Domains and Unique Factorization of Ideals 39
3.1 Dedekind Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Factoring Primes 47
4.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1.1 Geometric Intuition . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 A Method for Factoring Primes that Often Works . . . . . . . . . . 51
4.3 A General Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.1 Inessential Discriminant Divisors . . . . . . . . . . . . . . . . 54
4.3.2 Remarks on Ideal Factorization in General . . . . . . . . . . . 55
4.3.3 Finding a p-Maximal Order . . . . . . . . . . . . . . . . . . . 56
4.3.4 General Factorization Algorithm of Buchman-Lenstra . . . . 56
34 CONTENTS
5 The Chinese Remainder Theorem 59
5.1 The Chinese . . . . . . . . . . . . . . . . . . . . 59
5.1.1 CRT in the Integers . . . . . . . . . . . . . . . . . . . . . . . 59
5.1.2 CRT in General . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 Structural Applications of the CRT . . . . . . . . . . . . . . . . . . . 61
5.3 Computing Using the CRT . . . . . . . . . . . . . . . . . . . . . . . 63
5.3.1 Magma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3.2 PARI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6 Discrimants and Norms 67
6.1 ViewingO as a Lattice in a Real Vector Space . . . . . . . . . . . 67K
6.1.1 The Volume ofO . . . . . . . . . . . . . . . . . . . . . . . . 68K
6.2 Discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Norms of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7 Finiteness of the Class Group 75
7.1 The Class Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.2 Class Number 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.3 More About Computing Class Groups . . . . . . . . . . . . . . . . . 81
8 Dirichlet’s Unit Theorem 85
8.1 The Group of Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.2 Examples with Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
8.2.1 Pell’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 91
8.2.2 Examples with Various Signatures . . . . . . . . . . . . . . . 92
9 Decomposition and Inertia Groups 97
9.1 Galois Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
9.2 Decomposition of Primes: efg =n . . . . . . . . . . . . . . . . . . . 99
9.2.1 Quadratic Extensions . . . . . . . . . . . . . . . . . . . . . . 100
9.2.2 The Cube Root of Two . . . . . . . . . . . . . . . . . . . . . 101
9.3 The Decomposition Group . . . . . . . . . . . . . . . . . . . . . . . . 102
9.3.1 Galois groups of nite elds . . . . . . . . . . . . . . . . . . . 103
9.3.2 The Exact Sequence . . . . . . . . . . . . . . . . . . . . . . . 104
9.4 Frobenius Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
9.5 Galois Representations, L-series and a Conjecture of Artin . . . . . . 106
10 Elliptic Curves, Galois Representations, and L-functions 109
10.1 Groups Attached to Elliptic Curves . . . . . . . . . . . . . . . . . . . 109
10.1.1 Abelian Groups Attached to Elliptic Curves . . . . . . . . . . 110
10.1.2 A Formula for Adding Points . . . . . . . . . . . . . . . . . . 112
10.1.3 Other Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
10.2 Galois Representations Attached to Elliptic Curves . . . . . . . . . . 113
10.2.1 Modularity of Elliptic Curves over Q . . . . . . . . . . . . . . 115CONTENTS 5
11 Galois Cohomology 117
11.1 Group Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
11.1.1 Group Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
11.2 Modules and Group Cohomology . . . . . . . . . . . . . . . . . . . . 117
11.2.1 Example Application of the Theorem . . . . . . . . . . . . . . 119
11.3 In ation and Restriction . . . . . . . . . . . . . . . . . . . . . . . . . 120
11.4 Galois Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
12 The Weak Mordell-Weil Theorem 123
12.1 Kummer Theory of Number Fields . . . . . . . . . . . . . . . . . . . 123
12.2 Proof of the Weak Mordell-Weil Theorem . . . . . . . . . . . . . . . 125
II Adelic Viewpoint 129
13 Valuations 131
13.1 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
13.2 Types of Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
13.3 Examples of Valuations . . . . . . . . . . . . . . . . . . . . . . . . . 137
14 Topology and Completeness 141
14.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
14.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
14.2.1 p-adic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 144
14.2.2 The Field of p-adic Numbers . . . . . . . . . . . . . . . . . . 147
14.2.3 The Topology of Q (is Weird) . . . . . . . . . . . . . . . . . 148N
14.2.4 The Local-to-Global Principle of Hasse and Minkowski . . . . 149
14.3 Weak Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
15 Adic Numbers: The Finite Residue Field Case 153
15.1 Finite Residue Field Case . . . . . . . . . . . . . . . . . . . . . . . . 153
16 Normed Spaces and Tensor Products 161
16.1 Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
16.2 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
17 Extensions and Normalizations of Valuations 169
17.1 of Valuations . . . . . . . . . . . . . . . . . . . . . . . . . 169
17.2 Extensions of Normalized Valuations . . . . . . . . . . . . . . . . . . 174
18 Global Fields and Adeles 177
18.1 Global Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
18.2 Restricted Topological Products . . . . . . . . . . . . . . . . . . . . . 181
18.3 The Adele Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
18.4 Strong Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 1866 CONTENTS
19 Ideles and Ideals 191
19.1 The Idele Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
19.2 Ideals and Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
19.2.1 The Function Field Case . . . . . . . . . . . . . . . . . . . . . 196
19.2.2 Jacobians of Curves . . . . . . . . . . . . . . . . . . . . . . . 196
20 Exercises 197Preface
This book is based on notes the author created for a one-semester undergraduate
course on Algebraic Number Theory, which the author taught at Harvard during
Spring 2004 and Spring 2005. This book was mainly inspired by the [SD01, Ch. 1]
and Cassels’s article Global Fields in [Cas67]
|||||||||
- Copyright: William Stein, 2005, 2007.
License: Creative Commons Attribution-Share Alike 3.0 License
Please send any typos or corrections to wstein@gmail.com.
78 CONTENTS
Acknowledgement: This book closely builds on Swinnerton-Dyer’s book [SD01]
and Cassels’s article [Cas67]. Many of the students of Math 129 at Harvard dur-
ing Spring 2004 and 2005 made helpful comments: Jennifer Balakrishnan, Peter
Behrooz, Jonathan Bloom, David Escott Jayce Getz, Michael Hamburg, Deniz Ku-
ral, Danielle li, Andrew Ostergaard, Gregory Price, Grant Schoenebeck, Jennifer
Sinnott, Stephen Walker, Daniel Weissman, and Inna Zakharevich in 2004; Mauro
Braunstein, Steven Byrnes, William Fithian, Frank Kelly, Alison Miller, Nizamed-
din Ordulu, Corina Patrascu, Anatoly Preygel, Emily Riehl, Gary Sivek, Steven
Sivek, Kaloyan Slavov, Gregory Valiant, and Yan Zhang in 2005. Also the course
assistants Matt Bainbridge and Andrei Jorza made many helpful comments. The
+mathemtical software [S 10], [PAR], and [BCP97] were used in writing this book.
This material is based upon work supported by the National Science Foundation
under Grant No. 0400386.Chapter 1
Introduction
1.1 Mathematical background
In addition to general mathematical maturity, this book assumes you have the
following background:
Basics of nite group theory
Commutative rings, ideals, quotient rings
Some elementary number theory
Basic Galois theory of elds
Point set topology
Basic of topological rings, groups, and measure theory
For example, if you have never worked with nite groups before, you should read
another book rst. If you haven’t seen much elementary ring theory, there is still
hope, but you will have to do some additional reading and exercises. We will brie y
review the basics of the Galois theory of number elds.
Some of the homework problems involve using a computer, but there are ex-
amples which you can build on. We will not assume that you have a program-
ming background or know much about algorithms. Most of the book uses Sage
http://sagemath.org, which is free open source mathematical software. The fol-
lowing is an example Sage session:
sage: 2 + 2
4
sage: k.<a> = NumberField(x^2 + 1); k
Number Field in a with defining polynomial x^2 + 1
1.2 What is algebraic number theory?
A number eld K is a nite degree algebraic extension of the rational numbers Q.
The primitive element theorem from Galois theory asserts that every such extension
910 CHAPTER 1. INTRODUCTION
can be represented as the set of all polynomials of degree at most d = [K : Q] =
dim K in a single algebraic number :Q
( )
mX
nK = Q() = a :a 2 Q :n n
n=0
Here is a root of a polynomial with coe cients in Q.
Algebraic number theory involves using techniques from (mostly commutative)
algebra and nite group theory to gain a deeper understanding of the arithmetic
of number elds and related objects (e.g., functions elds, elliptic curves, etc.).
The main objects that we study in this book are number elds, rings of integers of
number elds, unit groups, ideal class groups, norms, traces, discriminants, prime
ideals, Hilbert and other class elds and associated reciprocity laws, zeta and L-
functions, and algorithms for computing each of the above.
1.2.1 Topics in this book
These are some of the main topics that are discussed in this book:
Rings of integers of number elds
Unique factorization of ideals in Dedekind domains
Structure of the group of units of the ring of integers
Finiteness of the group of equivalence classes of ideals of the ring of integers
(the \class group")
Decomposition and inertia groups, Frobenius elements
Rami cation
Discriminant and di erent
Quadratic and biquadratic elds
Cyclotomic elds (and applications)
How to use a computer to compute with many of the above objects (both
algorithms and actual use of software).
Valuations on elds
Completions (p-adic elds)
Adeles and Ideles
Note that we will not do anything nontrivial with zeta functions or L-functions.
1.3 Some applications of algebraic number theory
The following examples illustrate that learning algebraic number theory as soon as
possible is an excellent investment of your time.