The theory of algebraic numbers:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Washington, DC
Math. Assoc. of America
1975
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | The Carus mathematical monographs
9 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 159 |
| Beschreibung: | 162 S. |
| ISBN: | 0883850184 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804136070464405504 |
|---|---|
| adam_text | CONTENTS
CHAPTER
I. Divisibility
1. Uniqueness of factorization 1
2. A general problem 5
3. The Gaussian integers 7
Problems 11
II. The Gaussian Primes
1. Rational and Gaussian primes 14
2. Congruences 14
3. Determination of the Gaussian primes 18
4. Fermat s theorem for Gaussian primes 21
Problems 23
III. Polynomials over a field
1. The ring of polynomials 25
2. The Eisenstein irreducibility criterion 30
3. Symmetric polynomials 35
Problems 40
IV. Algebraic Number Fields
1. Numbers algebraic over a field 44
2. Extensions of a field 46
3. Algebraic and transcendental numbers 51
Problems 56
V. Bases
1. Bases and finite extensions 59
2. Properties of finite extensions 62
3. Conjugates and discriminants 64
4. The cyclotomic field 68
Problems 70
VI. Algebraic Integers and Integral Bases
1. Algebraic integers 74
2. The integers in a quadratic field 77
3. Integral bases 79
4. Examples of integral bases 82
Problems 86
CONTENTS
CHAPTER
VII. Arithmetic in Algebraic Number Fields
1. Units and primes 88
2. Units in a quadratic field 90
3. The uniqueness of factorization 93
4. Ideals in an algebraic number field 95
Problems 98
VIII. The Fundamental Theorem of Ideal Theory
1. Basic properties of ideals 102
2. The classical proof of the unique factorization
theorem 107
3. The modern proof 112
Problems 116
IX. Consequences of the Fundamental Theorem
1. The highest common factor of two ideals 120
2. Unique factorization of integers 123
3. The problem of ramification 125
4. Congruences and norms 128
5. Further properties of norms 132
Problems 135
X. Ideal Classes and Class Numbers
1. Ideal classes 139
2. Class numbers 140
Problems 144
XI. The Fermat Conjecture
1. Pythagorean triples 146
2. The Fermat conjecture 147
3. Units in cyclotomic fields 150
4. Rummer s theorem 155
Problems 158
References 159
List of Symbols 160
Index 161
|
| adam_txt |
CONTENTS
CHAPTER
I. Divisibility
1. Uniqueness of factorization 1
2. A general problem 5
3. The Gaussian integers 7
Problems 11
II. The Gaussian Primes
1. Rational and Gaussian primes 14
2. Congruences 14
3. Determination of the Gaussian primes 18
4. Fermat's theorem for Gaussian primes 21
Problems 23
III. Polynomials over a field
1. The ring of polynomials 25
2. The Eisenstein irreducibility criterion 30
3. Symmetric polynomials 35
Problems 40
IV. Algebraic Number Fields
1. Numbers algebraic over a field 44
2. Extensions of a field 46
3. Algebraic and transcendental numbers 51
Problems 56
V. Bases
1. Bases and finite extensions 59
2. Properties of finite extensions 62
3. Conjugates and discriminants 64
4. The cyclotomic field 68
Problems 70
VI. Algebraic Integers and Integral Bases
1. Algebraic integers 74
2. The integers in a quadratic field 77
3. Integral bases 79
4. Examples of integral bases 82
Problems 86
CONTENTS
CHAPTER
VII. Arithmetic in Algebraic Number Fields
1. Units and primes 88
2. Units in a quadratic field 90
3. The uniqueness of factorization 93
4. Ideals in an algebraic number field 95
Problems 98
VIII. The Fundamental Theorem of Ideal Theory
1. Basic properties of ideals 102
2. The classical proof of the unique factorization
theorem 107
3. The modern proof 112
Problems 116
IX. Consequences of the Fundamental Theorem
1. The highest common factor of two ideals 120
2. Unique factorization of integers 123
3. The problem of ramification 125
4. Congruences and norms 128
5. Further properties of norms 132
Problems 135
X. Ideal Classes and Class Numbers
1. Ideal classes 139
2. Class numbers 140
Problems 144
XI. The Fermat Conjecture
1. Pythagorean triples 146
2. The Fermat conjecture 147
3. Units in cyclotomic fields 150
4. Rummer's theorem 155
Problems 158
References 159
List of Symbols 160
Index 161 |
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| author | Pollard, Harry 1919-1985 Diamond, Harold G. 1940- |
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| classification_rvk | SK 180 |
| ctrlnum | (OCoLC)2212114 (DE-599)BVBBV022095412 |
| dewey-full | 512/.74 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed. |
| format | Book |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:14:56Z |
| indexdate | 2024-07-09T20:50:33Z |
| institution | BVB |
| isbn | 0883850184 |
| language | English |
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| physical | 162 S. |
| publishDate | 1975 |
| publishDateSearch | 1975 |
| publishDateSort | 1975 |
| publisher | Math. Assoc. of America |
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| series | The Carus mathematical monographs |
| series2 | The Carus mathematical monographs |
| spelling | Pollard, Harry 1919-1985 Verfasser (DE-588)127083162 aut The theory of algebraic numbers by Harry Pollard and Harold G. Diamond 2. ed. Washington, DC Math. Assoc. of America 1975 162 S. txt rdacontent n rdamedia nc rdacarrier The Carus mathematical monographs 9 Literaturverz. S. 159 Nombres algébriques, Théorie des nombre algébrique inriac théorie nombre algébrique inriac Algebraic number theory Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 s DE-604 Zahlentheorie (DE-588)4067277-3 s 1\p DE-604 Diamond, Harold G. 1940- Verfasser (DE-588)142256420 aut The Carus mathematical monographs 9 (DE-604)BV001887236 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015310264&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Pollard, Harry 1919-1985 Diamond, Harold G. 1940- The theory of algebraic numbers The Carus mathematical monographs Nombres algébriques, Théorie des nombre algébrique inriac théorie nombre algébrique inriac Algebraic number theory Algebraische Zahlentheorie (DE-588)4001170-7 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4001170-7 (DE-588)4067277-3 |
| title | The theory of algebraic numbers |
| title_auth | The theory of algebraic numbers |
| title_exact_search | The theory of algebraic numbers |
| title_exact_search_txtP | The theory of algebraic numbers |
| title_full | The theory of algebraic numbers by Harry Pollard and Harold G. Diamond |
| title_fullStr | The theory of algebraic numbers by Harry Pollard and Harold G. Diamond |
| title_full_unstemmed | The theory of algebraic numbers by Harry Pollard and Harold G. Diamond |
| title_short | The theory of algebraic numbers |
| title_sort | the theory of algebraic numbers |
| topic | Nombres algébriques, Théorie des nombre algébrique inriac théorie nombre algébrique inriac Algebraic number theory Algebraische Zahlentheorie (DE-588)4001170-7 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Nombres algébriques, Théorie des nombre algébrique théorie nombre algébrique Algebraic number theory Algebraische Zahlentheorie Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015310264&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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