Number theory: an introduction to mathematics
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2009
|
| Edition: | 2. ed. |
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 610 S. graph. Darst. |
| ISBN: | 9780387894850 |
Staff View
MARC
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| 245 | 1 | 0 | |a Number theory |b an introduction to mathematics |c W. A. Coppel |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York, NY |b Springer |c 2009 | |
| 300 | |a XIII, 610 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Universitext | |
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Record in the Search Index
| _version_ | 1804139211791532032 |
|---|---|
| adam_text | Contents
Preface
to the Second Edition
....................................... xi
Part A
I The Expanding Universe of Numbers
............................ 1
0
Sets, Relations and Mappings
................................. 1
1
Natural Numbers
............................................ 5
2
Integers and Rational Numbers
................................ 10
3
Real Numbers
.............................................. 17
4
Metric Spaces
.............................................. 27
5
Complex Numbers
.......................................... 39
6
Quaternions and Octonions
................................... 48
7
Groups
.................................................... 55
8
Rings and Fields
............................................ 60
9
Vector Spaces and Associative Algebras
........................ 64
10
Inner Product Spaces
........................................ 71
11
Further Remarks
............................................ 75
12
Selected References
......................................... 79
Additional References
............................................ 82
II Divisibility
.................................................. 83
1
Greatest Common Divisors
................................... 83
2
The
Bézout
Identity
.......................................... 90
3
Polynomials
................................................ 96
4
Euclidean Domains
.......................................... 104
5
Congruences
............................................... 106
6
Sums of Squares
............................................ 119
7
Further Remarks
............................................ 123
8
Selected References
......................................... 126
Additional References
............................................
1
27
viii Contents
III
More on Divisibility
.......................................... 129
1
The Law of Quadratic Reciprocity
.............................129
2
Quadratic Fields
............................................140
3
Multiplicative Functions
......................................152
4
Linear Diophantine Equations
.................................161
5
Further Remarks
............................................174
6
Selected References
.........................................176
Additional References
............................................178
IV Continued Fractions and Their Uses
............................ 179
1
The Continued Fraction Algorithm
.............................179
2
Diophantine Approximation
...................................185
3
Periodic Continued Fractions
..................................191
4
Quadratic Diophantine Equations
..............................195
5
The Modular Group
.........................................201
6
Non-Euclidean Geometry
.....................................208
7
Complements
...............................................211
8
Further Remarks
............................................217
9
Selected References
.........................................220
Additional References
............................................222
V Hadamard s Determinant Problem
............................. 223
1
What is a Determinant?
......................................223
2
Hadamard
Matrices
..........................................229
3
The Art of Weighing
.........................................233
4
Some Matrix Theory
.........................................237
5
Application to Hadamard s Determinant Problem
.................243
6
Designs
....................................................247
7
Groups and Codes
...........................................251
8
Further Remarks
............................................256
9
Selected References
.........................................258
VI Hensel s/r-adic Numbers
...................................... 261
1
Valued Fields
...............................................261
2
Equivalence
................................................265
3
Completions
................................................268
4
Non-Archimedean Valued Fields
...............................273
5
Hensel s Lemma
............................................277
6
Locally Compact Valued Fields
................................284
7
Further Remarks
............................................290
8
Selected References
.........................................290
Contents ix
Part
В
VII
The Arithmetic of Quadratic Forms
............................. 291
1
Quadratic Spaces
............................................291
2
The Hubert Symbol
.........................................303
3
The Hasse-Minkowski Theorem
...............................312
4
Supplements
...............................................322
5
Further Remarks
............................................324
6
Selected References
.........................................325
VIII
The Geometry of Numbers
.................................... 327
1
Minkowski s Lattice Point Theorem
............................327
2
Lattices
....................................................330
3
Proof of the Lattice Point Theorem; Other Results
................334
4
Voronoi Cells
...............................................342
5
Densest Packings
............................................347
6
Mahler s Compactness Theorem
...............................352
7
Further Remarks
............................................357
8
Selected References
.........................................360
Additional References
............................................362
IX The Number of Prime Numbers
................................ 363
1
Finding the Problem
.........................................363
2
Chebyshev s Functions
.......................................367
3
Proof of the Prime Number Theorem
...........................370
4
The Riemann Hypothesis
.....................................377
5
Generalizations and Analogues
................................384
6
Alternative Formulations
.....................................389
7
Some Further Problems
......................................392
8
Further Remarks
............................................394
9
Selected References
.........................................395
Additional References
............................................398
X A Character Study
........................................... 399
1
Primes in Arithmetic Progressions
.............................399
2
Characters of Finite Abelian Groups
............................400
3
Proof of the Prime Number Theorem for Arithmetic Progressions
... 403
4
Representations of Arbitrary Finite Groups
......................410
5
Characters of Arbitrary Finite Groups
..........................414
6
Induced Representations and Examples
.........................419
7
Applications
................................................425
8
Generalizations
.............................................432
9
Further Remarks
............................................443
10
Selected References
.........................................444
x
Contents
XI
Uniform Distribution
and Ergodic Theory
....................... 447
1 Uniform Distribution ........................................447
2
Discrepancy
................................................459
3 Birkhoff s Ergodic Theorem..................................464
4 Applications................................................472
5
Recurrence
.................................................483
6
Further Remarks
............................................488
7
Selected References
.........................................490
Additional Reference
.............................................492
XII
Elliptic Functions
............................................ 493
1
Elliptic Integrals
............................................493
2
The Arithmetic-Geometric Mean
..............................502
3
Elliptic Functions
...........................................509
4
Theta Functions
.............................................517
5
Jacobian Elliptic Functions
...................................525
6
The Modular Function
.......................................531
7
Further Remarks
............................................536
8
Selected References
.........................................539
XIII
Connections with Number Theory
.............................. 541
1
Sums of Squares
............................................541
2
Partitions
..................................................544
3
Cubic Curves
...............................................549
4
Mordelľs
Theorem
..........................................558
5
Further Results and Conjectures
...............................569
6
Some Applications
..........................................575
7
Further Remarks
............................................581
8
Selected References
.........................................584
Additional References
............................................586
Notations
........................................................ 587
Axioms
.......................................................... 591
Index
............................................................ 592
|
| any_adam_object | 1 |
| author | Coppel, William A. |
| author_GND | (DE-588)108974839 |
| author_facet | Coppel, William A. |
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| dewey-full | 512.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.7 |
| dewey-search | 512.7 |
| dewey-sort | 3512.7 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| isbn | 9780387894850 |
| language | English |
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| physical | XIII, 610 S. graph. Darst. |
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| spelling | Coppel, William A. Verfasser (DE-588)108974839 aut Number theory an introduction to mathematics W. A. Coppel 2. ed. New York, NY Springer 2009 XIII, 610 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Nombres, Théorie des ram Number theory Zahlentheorie (DE-588)4067277-3 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Zahlentheorie (DE-588)4067277-3 s DE-604 Erscheint auch als Online-Ausgabe 978-0-387-89486-7 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017618061&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Coppel, William A. Number theory an introduction to mathematics Nombres, Théorie des ram Number theory Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4123623-3 |
| title | Number theory an introduction to mathematics |
| title_auth | Number theory an introduction to mathematics |
| title_exact_search | Number theory an introduction to mathematics |
| title_full | Number theory an introduction to mathematics W. A. Coppel |
| title_fullStr | Number theory an introduction to mathematics W. A. Coppel |
| title_full_unstemmed | Number theory an introduction to mathematics W. A. Coppel |
| title_short | Number theory |
| title_sort | number theory an introduction to mathematics |
| title_sub | an introduction to mathematics |
| topic | Nombres, Théorie des ram Number theory Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Nombres, Théorie des Number theory Zahlentheorie Lehrbuch |
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