Topics from the theory of numbers:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston, Basel, Berlin
Birkhäuser
2009
|
| Edition: | 2. ed., reprint of the 1984 2. ed. |
| Series: | Modern Birkhäuser Classics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 333 S. Illustrationen |
| ISBN: | 9780817648374 |
Staff View
MARC
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| 020 | |a 9780817648374 |9 978-0-8176-4837-4 | ||
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| 100 | 1 | |a Grosswald, Emil |d 1912-1989 |e Verfasser |0 (DE-588)119267551 |4 aut | |
| 245 | 1 | 0 | |a Topics from the theory of numbers |c Emil Grosswald |
| 250 | |a 2. ed., reprint of the 1984 2. ed. | ||
| 264 | 1 | |a Boston, Basel, Berlin |b Birkhäuser |c 2009 | |
| 300 | |a 333 S. |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Modern Birkhäuser Classics | |
| 650 | 4 | |a Números, Teoría de | |
| 650 | 0 | 7 | |a Zahlentheorie |0 (DE-588)4067277-3 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-017071917 | ||
Record in the Search Index
| _version_ | 1804138551779000320 |
|---|---|
| adam_text | Contents
Preface
xiii
Preface
to the First Edition
xv
PART ONE INTRODUCTION, HISTORICAL
BACKGROUND, AND NOTATIONS
Chapter
1
Introduction and Historical Background
3
1.
Number Theory—The Queen of Mathematics
3
2.
A Problem
4
3.
Something About the Contents of This Book
6
4.
Number Theory and Other Branches of Mathematics
6
5.
The Distribution of Prime Numbers
7
6.
Fermat s Last Theorem
9
7.
The Theory of Partitions
10
8.
Elementary Number Theory
11
Chapter
2
Introductory Remarks and Notations
13
PART TWO ELEMENTARY NUMBER THEORY
Chapter
3
Divisibility
19
1.
Generalities and Fundamental Theorem
19
2.
Discussion of Two Objections
25
3.
An Example of Hubert
26
4.
Two Further Theorems
28
5.
Some Results Concerning the Distribution of Prime Numbers
28
Chapter
4
Congruences
35
1.
Congruences as Equivalence Relations—General Properties
35
2.
Operations with Residue Classes
39
3.
Theorems of
Fermat,
Euler,
and Wilson
43
4.
Linear Congruences
45
5.
The Chinese Remainder Theorem
48
6.
On Primitive Roots
50
7.
♦Congruences of Higher Degrees
55
Chapter
5
Quadratic Residues
67
1.
Introduction
67
2.
The Legendre Symbol and the Law of Quadratic Reciprocity
68
3.
*The Jacobi and
Kronecker
Symbols
74
Chapter
6
Arithmetical Functions
81
1.
Introduction
81
2.
The Function [x]
81
3.
The Function
y
=
((χ))
83
4.
The
Euler
Function
φ {η)
85
5.
The
Möbius
Function
μ.(η)
87
6.
Liouville s Function
92
91
7.
The Function
òk(n)
8.
♦Perfect Numbers ^
9.
*Ramanujan Sums
:*
10.
Functions Related to Prime Numbers 97
11.
Formulae that Yield Primes 101
12.
**On the Sum Function M(x) 102
Chapter
7
The Theory of Partitions 109
1.
Introduction 109
2.
Definition and Notations 109
3.
Survey of Methods 112
4.
Generating Functions
*
*2
5.
Graphs m
6.
The Size of p(n) 127
7.
Some Lemmas 13°
8.
Proof of the Theorem 132
PART THREE TOPICS FROM ANALYTIC AND
ALGEBRAIC NUMBER THEORY
Chapter
8
The Distribution of Primes and the Riernann
Zeta
Function
143
1.
The Distribution of Primes and the Sieve Method 143
2.
From Tchebycheff to Landau 144
3.
The Riemann
Zeta
Function I45
4.
The
Zeta
Function and
чг(дг) 148
5.
Further Theory of the
Zeta
Function
149
6.
The Functional Equation
153
7.
The values of
ζ
(s)
at Integral Arguments
155
8.
The
Zeta
Function and Its Derivatives for
σ
Close to
1 158
9.
Comments on the
Zeta
Function
164
Chapter
9
The Prime Number Theorem
169
1. Introduction
169
2.
Sketch of the Proof
170
3.
Some Lemmas
173
4.
Proof of the PNT
177
Chapter
10
The Arithmetic of Number Fields
187
1.
Introduction
187
2.
Fields and Rings of Algebraic Numbers
189
3.
The Quadratic Field Q(V2)
192
4.
The Ring
Да/Г)
of Integers of
Κ- β(νΐ)
and its Units
195
5.
Factorization in
К
=
Q( vf)
197
6.
The Euclidean Algorithm in
ß(V2) 199
7.
The Field Q(4^b)
200
8.
The Field Q(V5)
204
9.
The Gaussian Field
К
= ß(
V^ì
) 205
10.
Quadratic Fields
207
11.
Cyclotomic Fields
208
12.
Other Algebraic Number Fields
211
13.
Integer and Units in Algebraic Number Fields
212
Chapter
11
Ideal Theory
219
1.
Introduction
219
2.
Definitions and Elementary Properties of Ideals
219
3.
Divisibility Properties of Ideals
222
4.
Uniqueness of Factorization of Ideals into Prime Ideals
224
5.
Ideal Classes and the Class Number
226
Chapter
12
Primes in Arithmetic Progressions
233
1.
Introduction and Dirichlet s Theorem
233
2.
Characters
235
3.
Dirichlet s ¿-Functions
238
4.
Proof of Theorem
1 243
5.
Some Auxiliary Results
243
6.
End of the Proof of Theorem
1 247
7.
Dirichlet s Approach
248
8.
Concluding Remarks
249
Chapter
13
Diophantine Equations
255
1.
Introduction
255
2.
Hubert s Tenth Problem
257
3.
Effective Methods
258
4.
The Genus of a Curve
259
5.
The Theorems and Conjectures of A. Weil
262
6.
The
Hasse
Principle
263
7.
Linear and Quadratic Diophantine Equations
264
8.
Representations by Sums of Two Squares
265
9.
Representations by Sums of Three Squares
267
10.
A Theorem of Legendre
269
11.
Proof of Legendre s Theorem
270
12.
Lagrange s Theorem and the Representation of Integers as
Sums of Four Squares
273
13.
Representations of Integers as Sums of
r
Squares
277
14.
The Mordell-Weil Theorem
277
15.
Bachet s Equation
281
16.
Final Remarks
285
Chapter
14
Fermat s Equation
289
1.
Introduction
289
2.
Solution of Fermat s Equation for
я
= 2 291
3.
Proof of the
FC
for
я
= 4 292
4.
Proof of the
FC
for
ρ
= 3 293
5.
Case I for
ρ
= 5 298
6.
Case I for an Arbitrary Prime
ρ
300
7.
Regular Primes
304
8.
Proof of the
FC
for Regular Primes in Case I
304
9.
Proof of the
FC
for Regular Primes in Case II
310
10.
Some Final Remarks
315
Answers to Selected Problems
321
Subject Index
327
Name Index
331
Index of Functions
335
|
| any_adam_object | 1 |
| author | Grosswald, Emil 1912-1989 |
| author_GND | (DE-588)119267551 |
| author_facet | Grosswald, Emil 1912-1989 |
| author_role | aut |
| author_sort | Grosswald, Emil 1912-1989 |
| author_variant | e g eg |
| building | Verbundindex |
| bvnumber | BV035266473 |
| classification_rvk | SK 180 |
| classification_tum | MAT 120f MAT 100f |
| ctrlnum | (OCoLC)630844969 (DE-599)BVBBV035266473 |
| discipline | Mathematik |
| edition | 2. ed., reprint of the 1984 2. ed. |
| format | Book |
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| id | DE-604.BV035266473 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:30:00Z |
| institution | BVB |
| isbn | 9780817648374 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017071917 |
| oclc_num | 630844969 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-11 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-11 DE-188 |
| physical | 333 S. Illustrationen |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Modern Birkhäuser Classics |
| spelling | Grosswald, Emil 1912-1989 Verfasser (DE-588)119267551 aut Topics from the theory of numbers Emil Grosswald 2. ed., reprint of the 1984 2. ed. Boston, Basel, Berlin Birkhäuser 2009 333 S. Illustrationen txt rdacontent n rdamedia nc rdacarrier Modern Birkhäuser Classics Números, Teoría de Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017071917&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Grosswald, Emil 1912-1989 Topics from the theory of numbers Números, Teoría de Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4067277-3 |
| title | Topics from the theory of numbers |
| title_auth | Topics from the theory of numbers |
| title_exact_search | Topics from the theory of numbers |
| title_full | Topics from the theory of numbers Emil Grosswald |
| title_fullStr | Topics from the theory of numbers Emil Grosswald |
| title_full_unstemmed | Topics from the theory of numbers Emil Grosswald |
| title_short | Topics from the theory of numbers |
| title_sort | topics from the theory of numbers |
| topic | Números, Teoría de Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Números, Teoría de Zahlentheorie |
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