Algebra and number theory: a selection of highlights
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Boston
De Gruyter
[2017]
|
| Series: | De Gruyter Textbook
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Erster Band eines 2-bändigen Sets ohne Zählung |
| Physical Description: | X, 332 Seiten Diagramme |
| ISBN: | 9783110515848 9783110523706 |
Staff View
MARC
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| 100 | 1 | |a Fine, Benjamin |d 1948- |e Verfasser |0 (DE-588)132345048 |4 aut | |
| 245 | 1 | 0 | |a Algebra and number theory |b a selection of highlights |c Benjamin Fine, Anthony Gaglione, Anja Moldenhauer, Gerhard Rosenberger, Dennis Spellman |
| 264 | 1 | |a Berlin ; Boston |b De Gruyter |c [2017] | |
| 264 | 4 | |c © 2017 | |
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| 653 | |a Fachhochschul-/Hochschulausbildung | ||
| 653 | |a Elementare Zahlentheorie | ||
| 653 | |a Galois-Theorie | ||
| 653 | |a Primzahl | ||
| 653 | |a Polynom | ||
| 653 | |a Zahlensystem | ||
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Record in the Search Index
| _version_ | 1804177837473660928 |
|---|---|
| adam_text | CONTENTS
PREFACE
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
3
3.1
3.2
3.3
3.4
3.5
4
4.1
4.1.1
4
.
1.2
4.1.3
4.1.4
4.2
4.3
5
5.1
5.2
5.3
THE NATURAL, INTEGRAL AND RATIONAL NUMBERS * 1
NUMBER THEORY AND AXIOMATIC SYSTEMS * 1
THE NATURAL NUMBERS AND INDUCTION * 1
THE INTEGERS
Z
* 10
THE RATIONAL NUMBERS Q * 13
THE ABSOLUTE VALUE IN
N, Z
AND Q * 15
DIVISION AND FACTORIZATION IN THE INTEGERS * 19
THE FUNDAMENTAL THEOREM OF ARITHMETIC * 19
THE DIVISION ALGORITHM AND THE GREATEST COMMON DIVISOR * 23
THE EUCLIDEAN ALGORITHM * 26
LEAST COMMON MULTIPLES * 30
GENERAL GCD S AND LEM S * 33
MODULAR ARITHMETIC * 39
THE RING OF INTEGERS MODULO N * 39
UNITS AND THE EULER ^-FUNCTION * 43
RSA CRYPTOSYSTEM * 46
THE CHINESE REMAINDER THEOREM * 47
QUADRATIC RESIDUES * 54
EXCEPTIONAL NUMBERS * 61
THE FIBONACCI NUMBERS * 61
THE GOLDEN RECTANGLE * 67
SQUARES IN SEMICIRCLES * 68
SIDE LENGTH OF A REGULAR 10-GON * 69
CONSTRUCTION OF THE GOLDEN SECTION A WITH COMPASS AND STRAIGHTEDGE
FROM A GIVEN A E R, A 0 * 70
PERFECT NUMBERS AND MERSENNE NUMBERS * 71
FERMAT NUMBERS * 78
PYTHAGOREAN TRIPLES AND SUMS OF SQUARES * 83
THE PYTHAGOREAN THEOREM * 83
CLASSIFICATION OF THE PYTHAGOREAN TRIPLES * 85
SUM OF SQUARES * 89
6
6.1
6.2
6.3
6.3.1
6.3.2
6.4
6.5
6.5.1
6
.
5.2
6.5.3
6.6
6
.
6.1
6
.
6.2
6.6.3
7
7.1
7.2
7.3
7.3.1
7.4
8
8.1
8.2
8
.
2.1
8
.
2.2
8.3
9
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.7.1
9.7.2
9.7.3
POLYNOMIALS AND UNIQUE FACTORIZATION * 95
POLYNOMIALS OVER A RING * 95
DIVISIBILITY IN RINGS
----
98
THE RING OF POLYNOMIALS OVER A FIELD K * 100
THE DIVISION ALGORITHM FOR POLYNOMIALS * 101
ZEROS OF POLYNOMIALS
----
103
HORNER-SCHEME
----
108
THE EUCLIDEAN ALGORITHM AND GREATEST COMMON DIVISOR OF POLYNOMIALS
OVER FIELDS * 112
THE EUCLIDEAN ALGORITHM FORK[X]
----
114
UNIQUE FACTORIZATION OF POLYNOMIALS IN K[X] * 115
GENERAL UNIQUE FACTORIZATION DOMAINS * 116
POLYNOMIAL INTERPOLATION AND THE SHAMIR SECRET SHARING SCHEME * 117
SECRET SHARING
----
117
POLYNOMIAL INTERPOLATION OVER A FIELD
K * 117
THE SHAMIR SECRET SHARING SCHEME
----
121
FIELD EXTENSIONS AND SPLITTING FIELDS * 125
FIELDS, SUBFIELD AND CHARACTERISTIC * 125
FIELD EXTENSIONS
----
126
FINITE AND ALGEBRAIC FIELD EXTENSIONS * 131
FINITE FIELDS
----
134
SPLITTING FIELDS
----
135
PERMUTATIONS AND SYMMETRIC POLYNOMIALS * 141
PERMUTATIONS
----
141
CYCLE DECOMPOSITION OF A PERMUTATION * 144
CONJUGATE ELEMENTS IN SN * 147
MARSHALL HALL S THEOREM
----
148
SYMMETRIC POLYNOMIALS * 151
REAL NUMBERS * 157
THE REAL NUMBER SYSTEM * 157
DECIMAL REPRESENTATION OF REAL NUMBERS * 168
PERIODIC DECIMAL NUMBERS AND THE RATIONAL NUMBER * 172
THE UNCOUNTABILITY OF R
----
173
CONTINUED FRACTION REPRESENTATION OF REAL NUMBERS * 175
THEOREM OF DIRICHLET AND CAUCHY S INEQUALITY * 176
P-ADIC NUMBERS * 178
NORMED FIELDS AND CAUCHY COMPLETIONS * 179
THE P-ADIC FIELDS
----
180
THE P-ADIC NORM
----
183
9.7.4 THE CONSTRUCTION OF
QP
-----
184
9.7.5 OSTROWSKI*S THEOREM
----
185
10 THE COMPLEX NUMBERS, THE FUNDAMENTAL THEOREM OF ALGEBRA AND
POLYNOMIAL
EQUATIONS * 189
10.1 THE FIELD C OF COMPLEX NUMBERS * 189
10.2 THE COMPLEX PLANE * 193
10.2.1 GEOMETRIC INTERPRETATION OF COMPLEX OPERATIONS * 196
10.2.2 POLAR FORM AND EULER*S IDENTITY
-----
197
10.2.3 OTHER CONSTRUCTIONS OF C
-----
201
10.2.4 THE GAUSSIAN INTEGERS
----
201
10.3 THE FUNDAMENTAL THEOREM OF ALGEBRA * 202
10.3.1 FIRST PROOF OF THE FUNDAMENTAL THEOREM OF ALGEBRA * 204
10.3.2 SECOND PROOF OF THE FUNDAMENTAL THEOREM OF ALGEBRA
----
207
10.4 SOLVING POLYNOMIAL EQUATIONS IN TERMS OF RADICALS * 209
10.5 SKEW FIELD EXTENSIONS OF C AND FROBENIUS*S THEOREM
----
220
11 QUADRATIC NUMBER FIELDS AND PELL*S EQUATION * 227
11.1 ALGEBRAIC EXTENSIONS OF Q * 227
11.2 ALGEBRAIC AND TRANSCENDENTAL NUMBERS * 228
11.3 DISCRIMINANT AND NORM
-----
230
11.4 ALGEBRAIC INTEGERS * 235
11.4.1 THE RING OF ALGEBRAIC INTEGERS * 236
11.5 INTEGRAL BASES
----
238
11.6 QUADRATIC FIELDS AND QUADRATIC INTEGERS * 240
12 TRANSCENDENTAL NUMBERS AND THE NUMBERS
E
AND IR * 249
12.1 THE NUMBERS
E AND N * 249
12.1.1 CALCULATION E OF
N
----
251
12.2 THE IRRATIONALITY OF E AND N * 256
12.3 E AND N THROUGHOUT MATHEMATICS
----
263
12.3.1 THE NORMAL DISTRIBUTION
-----
263
12.3.2 THE GAMMA FUNCTION AND STIRLING*S APPROXIMATION * 264
12.3.3 THE WALLIS PRODUCT FORMULA
-----
266
12.4 EXISTENCE OF A TRANSCENDENTAL NUMBER * 270
12.5 THE TRANSCENDENCE OF E AND N * 273
12.6 AN AMAZING PROPERTY OF N AND A CONNECTION TO PRIME NUMBERS * 282
13 COMPASS AND STRAIGHTEDGE CONSTRUCTIONS AND THE CLASSICAL
PROBLEMS * 289
13.1 HISTORICAL REMARKS * 289
13.2 GEOMETRIC CONSTRUCTIONS
-----
289
13.3 FOUR CLASSICAL CONSTRUCTION PROBLEMS * 296
13.3.1 SQUARING THE CIRCLE (PROBLEM OF ANAXAGORAS 500-428 BC)
----
296
13.3.2 THE DOUBLING OF THE CUBE OR THE PROBLEM FROM DELI
----
296
13.3.3 THE TRISECTION OF AN ANGLE * 297
13.3.4 CONSTRUCTION OF A REGULAR N-GON * 298
14 EUCLIDEAN VECTOR SPACES * 303
14.1 LENGTH AND ANGLE
----
303
14.2 ORTHOGONALITY AND APPLICATIONS IN R2 AND IR3 * 309
14.3 ORTHONORMALIZATION AND CLOSEST VECTOR * 317
14.4 POLYNOMIAL APPROXIMATION
----
321
14.5 SECRET SHARING SCHEME USING THE CLOSEST VECTOR THEOREM * 323
BIBLIOGRAPHY
----
327
INDEX
----
329
|
| any_adam_object | 1 |
| author | Fine, Benjamin 1948- Gaglione, Anthony M. 1946- Moldenhauer, Anja Rosenberger, Gerhard 1944- Spellman, Dennis |
| author_GND | (DE-588)132345048 (DE-588)1060692333 (DE-588)1147606471 (DE-588)13155221X |
| author_facet | Fine, Benjamin 1948- Gaglione, Anthony M. 1946- Moldenhauer, Anja Rosenberger, Gerhard 1944- Spellman, Dennis |
| author_role | aut aut aut aut aut |
| author_sort | Fine, Benjamin 1948- |
| author_variant | b f bf a m g am amg a m am g r gr d s ds |
| building | Verbundindex |
| bvnumber | BV044492129 |
| classification_rvk | SK 180 |
| ctrlnum | (OCoLC)1002282019 (DE-599)DNB1130669823 |
| dewey-full | 510 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
| dewey-raw | 510 |
| dewey-search | 510 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV044492129 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:54:25Z |
| institution | BVB |
| institution_GND | (DE-588)10095502-2 |
| isbn | 9783110515848 9783110523706 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029892059 |
| oclc_num | 1002282019 |
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| owner_facet | DE-19 DE-BY-UBM DE-29T DE-634 DE-20 DE-83 DE-11 DE-188 |
| physical | X, 332 Seiten Diagramme |
| publishDate | 2017 |
| publishDateSearch | 2017 |
| publishDateSort | 2017 |
| publisher | De Gruyter |
| record_format | marc |
| series2 | De Gruyter Textbook |
| spelling | Fine, Benjamin 1948- Verfasser (DE-588)132345048 aut Algebra and number theory a selection of highlights Benjamin Fine, Anthony Gaglione, Anja Moldenhauer, Gerhard Rosenberger, Dennis Spellman Berlin ; Boston De Gruyter [2017] © 2017 X, 332 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier De Gruyter Textbook Erster Band eines 2-bändigen Sets ohne Zählung Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Algebra (DE-588)4001156-2 gnd rswk-swf Fachhochschul-/Hochschulausbildung Elementare Zahlentheorie Galois-Theorie Primzahl Polynom Zahlensystem Algebra (DE-588)4001156-2 s Zahlentheorie (DE-588)4067277-3 s DE-604 Gaglione, Anthony M. 1946- (DE-588)1060692333 aut Moldenhauer, Anja (DE-588)1147606471 aut Rosenberger, Gerhard 1944- (DE-588)13155221X aut Spellman, Dennis aut Walter de Gruyter GmbH & Co. KG (DE-588)10095502-2 pbl Erscheint auch als Online-Ausgabe, PDF 978-3-11-051614-2 Erscheint auch als Online-Ausgabe, EPUB 978-3-11-051626-5 Ergänzung zu Fine, Benjamin Geometry and discrete mathematics 978-3-11-052145-0 (DE-604)BV045263756 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029892059&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fine, Benjamin 1948- Gaglione, Anthony M. 1946- Moldenhauer, Anja Rosenberger, Gerhard 1944- Spellman, Dennis Algebra and number theory a selection of highlights Zahlentheorie (DE-588)4067277-3 gnd Algebra (DE-588)4001156-2 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4001156-2 |
| title | Algebra and number theory a selection of highlights |
| title_auth | Algebra and number theory a selection of highlights |
| title_exact_search | Algebra and number theory a selection of highlights |
| title_full | Algebra and number theory a selection of highlights Benjamin Fine, Anthony Gaglione, Anja Moldenhauer, Gerhard Rosenberger, Dennis Spellman |
| title_fullStr | Algebra and number theory a selection of highlights Benjamin Fine, Anthony Gaglione, Anja Moldenhauer, Gerhard Rosenberger, Dennis Spellman |
| title_full_unstemmed | Algebra and number theory a selection of highlights Benjamin Fine, Anthony Gaglione, Anja Moldenhauer, Gerhard Rosenberger, Dennis Spellman |
| title_short | Algebra and number theory |
| title_sort | algebra and number theory a selection of highlights |
| title_sub | a selection of highlights |
| topic | Zahlentheorie (DE-588)4067277-3 gnd Algebra (DE-588)4001156-2 gnd |
| topic_facet | Zahlentheorie Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029892059&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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