Algebraic theory of numbers: Translated from the French by Allan J. Silberger
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
London
Dover Publ.
2008
|
| Edition: | 1. Dover ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 109 S. |
| ISBN: | 0486466663 9780486466668 |
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MARC
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| 240 | 1 | 0 | |a Théorie algébrique des nombres [engl.] |
| 245 | 1 | 0 | |a Algebraic theory of numbers |b Translated from the French by Allan J. Silberger |
| 250 | |a 1. Dover ed. | ||
| 264 | 1 | |a London |b Dover Publ. |c 2008 | |
| 300 | |a 109 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Algebraic number theory | |
| 650 | 0 | 7 | |a Zahlentheorie |0 (DE-588)4067277-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Algebra |0 (DE-588)4001156-2 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804137310866898944 |
|---|---|
| adam_text | Contents
TRANSLATOR S
INTRODUCTION
........
,
7
INTRODUCTION
............9
NOTATIONS, DEFINITIONS, AND PREREQUISITES
. . . . . -.11
chapter
1
Principal ideal rings
. . . . . . . · · 13
1.
Divisibility in principal ideal rings
.......
jg
2.
An example: the diophantine equations X2
+
Y2
=
Z2 and X*
+
Y* = Z*
......_. . . · . 15
3.
Some lemmas concerning ideals; Euler s
çs-function
. . · . 17
4.
Some preliminaries concerning modules
. . . . · . 19
5.
Modules over principal ideal rings
.......21
6.
Roots of unity in afield
......... 23
7.
Finite fields
....-·····. 23
chapter
π
Elements integral over a ring; elements algebraic over a field
. 27
1.
Elements integral over a ring
........ 27
2.
Integrally closed rings
.........30
3.
Elements algebraic over a field. Algebraic extensions
. . · . 30
4.
Conjugate elements, conjugate fields
. . . · · . 32
5.
Integers in quadratic fields
........ 34
6.
Norms and traces
...···■■·. 36
7.
The discriminant
...······. 38
8.
The terminology of number fields
. . ..... 41
9.
Gyclotomic fields
...·····■. 42
Appendix The field of complex numbers is algebraically closed
. . . 44
chapter
m
Noeťherian
rings and Dedekind rings
. . . · . 46
1.
Noetherian rings and modules
. · ......
4g
2.
An application concerning integral elements
. . . . . 47
3.
Some preliminaries concerning ideals
.......47
4.
Dedekind rings
.........49
5.
The norm of an ideal
...·····. 52
6 CONTENTS
chapter rv Ideal classes and the unit theorem
...... 53
1.
Preliminaries concerning discrete subgroups of R*
. . . .53
2.
The canonical imbedding of a number field
..... 56
3.
Finiteness of the ideal class group
....... 57
4.
The unit theorem
.......... 59
5.
Units in imaginary quadratic fields
....... 62
6.
Units in real quadratic fields
........ 62
7.
A generalization of the unit theorem
....... 64
Appendix The calculation of a volume
....... 66
chapter
v
The splitting of prime ideals in an extension field
... 68
1.
Preliminaries concerning rings of fractions
...... 68
2.
The splitting of a prime ideal in an extension
..... 70
3.
The discriminant and ramification
....... 73
4.
The splitting of a prime number in a quadratic field
.... 76
5.
The quadratic reciprocity law
........ 77
6.
The two-squares theorem
......... 81
7.
The four-squares theorem
......... 82
chapter
vi
Galois extensions of number fields
...... 86
1.
Galois theory
........... 86
2.
The decomposition and inertia groups
...... 89
3.
The number field case. The Frobenius automorphism.
. . .91
4.
An application to cyclotomic fields
....... 92
5.
Another proof of the quadratic reciprocity law
..... 92
A SUPPLEMENT, WITHOUT PROOFS
......... 94
EXERCISES
............ 97
BIBLIOGRAPHY
............ 106
INDEX
............. 108
|
| adam_txt |
Contents
TRANSLATOR'S
INTRODUCTION
.
,
7
INTRODUCTION
.9
NOTATIONS, DEFINITIONS, AND PREREQUISITES
. . . . . -.11
chapter
1
Principal ideal rings
. . . . . . . · · 13
1.
Divisibility in principal ideal rings
.
jg
2.
An example: the diophantine equations X2
+
Y2
=
Z2 and X*
+
Y* = Z*
._. . . · . 15
3.
Some lemmas concerning ideals; Euler's
çs-function
. . · . 17
4.
Some preliminaries concerning modules
. . . . · . 19
5.
Modules over principal ideal rings
.21
6.
Roots of unity in afield
. 23
7.
Finite fields
.-·····. 23
chapter
π
Elements integral over a ring; elements algebraic over a field
. 27
1.
Elements integral over a ring
. 27
2.
Integrally closed rings
.30
3.
Elements algebraic over a field. Algebraic extensions
. . · . 30
4.
Conjugate elements, conjugate fields
. . . · · . 32
5.
Integers in quadratic fields
. 34
6.
Norms and traces
.···■■·. 36
7.
The discriminant
.······. 38
8.
The terminology of number fields
. . . 41
9.
Gyclotomic fields
.·····■. 42
Appendix The field of complex numbers is algebraically closed
. . . 44
chapter
m
Noeťherian
rings and Dedekind rings
. . . · . 46
1.
Noetherian rings and modules
. · .
4g
2.
An application concerning integral elements
. . . . . 47
3.
Some preliminaries concerning ideals
.47
4.
Dedekind rings
.49
5.
The norm of an ideal
.·····. 52
6 CONTENTS
chapter rv Ideal classes and the unit theorem
. 53
1.
Preliminaries concerning discrete subgroups of R*
. . . .53
2.
The canonical imbedding of a number field
. 56
3.
Finiteness of the ideal class group
. 57
4.
The unit theorem
. 59
5.
Units in imaginary quadratic fields
. 62
6.
Units in real quadratic fields
. 62
7.
A generalization of the unit theorem
. 64
Appendix The calculation of a volume
. 66
chapter
v
The splitting of prime ideals in an extension field
. 68
1.
Preliminaries concerning rings of fractions
. 68
2.
The splitting of a prime ideal in an extension
. 70
3.
The discriminant and ramification
. 73
4.
The splitting of a prime number in a quadratic field
. 76
5.
The quadratic reciprocity law
. 77
6.
The two-squares theorem
. 81
7.
The four-squares theorem
. 82
chapter
vi
Galois extensions of number fields
. 86
1.
Galois theory
. 86
2.
The decomposition and inertia groups
. 89
3.
The number field case. The Frobenius automorphism.
. . .91
4.
An application to cyclotomic fields
. 92
5.
Another proof of the quadratic reciprocity law
. 92
A SUPPLEMENT, WITHOUT PROOFS
. 94
EXERCISES
. 97
BIBLIOGRAPHY
. 106
INDEX
. 108 |
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| author | Samuel, Pierre 1921-2009 |
| author_GND | (DE-588)1089946481 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.7/4 |
| dewey-search | 512.7/4 |
| dewey-sort | 3512.7 14 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 1. Dover ed. |
| format | Book |
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| id | DE-604.BV023068764 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T19:32:06Z |
| indexdate | 2024-07-09T21:10:16Z |
| institution | BVB |
| isbn | 0486466663 9780486466668 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016271930 |
| oclc_num | 182779830 |
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| owner_facet | DE-355 DE-BY-UBR DE-19 DE-BY-UBM |
| physical | 109 S. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Dover Publ. |
| record_format | marc |
| spelling | Samuel, Pierre 1921-2009 Verfasser (DE-588)1089946481 aut Théorie algébrique des nombres [engl.] Algebraic theory of numbers Translated from the French by Allan J. Silberger 1. Dover ed. London Dover Publ. 2008 109 S. txt rdacontent n rdamedia nc rdacarrier Algebraic number theory Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Algebra (DE-588)4001156-2 gnd rswk-swf Algebra (DE-588)4001156-2 s 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=016271930&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Samuel, Pierre 1921-2009 Algebraic theory of numbers Translated from the French by Allan J. Silberger Algebraic number theory Zahlentheorie (DE-588)4067277-3 gnd Algebra (DE-588)4001156-2 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4001156-2 |
| title | Algebraic theory of numbers Translated from the French by Allan J. Silberger |
| title_alt | Théorie algébrique des nombres [engl.] |
| title_auth | Algebraic theory of numbers Translated from the French by Allan J. Silberger |
| title_exact_search | Algebraic theory of numbers Translated from the French by Allan J. Silberger |
| title_exact_search_txtP | Algebraic theory of numbers Translated from the French by Allan J. Silberger |
| title_full | Algebraic theory of numbers Translated from the French by Allan J. Silberger |
| title_fullStr | Algebraic theory of numbers Translated from the French by Allan J. Silberger |
| title_full_unstemmed | Algebraic theory of numbers Translated from the French by Allan J. Silberger |
| title_short | Algebraic theory of numbers |
| title_sort | algebraic theory of numbers translated from the french by allan j silberger |
| title_sub | Translated from the French by Allan J. Silberger |
| topic | Algebraic number theory Zahlentheorie (DE-588)4067277-3 gnd Algebra (DE-588)4001156-2 gnd |
| topic_facet | Algebraic number theory Zahlentheorie Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016271930&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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