Solving polynomial equation systems: 1 The Kronecker-Duval philosophy
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Cambridge University Press
2003
|
| Edition: | 1. publ. |
| Series: | Encyclopedia of mathematics and its applications
88 |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 423 S. |
| ISBN: | 0521811546 |
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Record in the Search Index
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| adam_text | Contents
Preface page xi
Part one: The Kronecker - Duval Philosophy 1
1 Euclid 3
1.1 The Division Algorithm 4
1.2 Euclidean Algorithm 6
L3 Bezout’s Identity and Extended Euclidean Algorithm 8
1.4 Roots of Polynomials 9
1.5 Factorization of Polynomials 10
1.6* Computing a gcd 12
1.6.1* Coefficient explosion 12
1.6.2* Modular Algorithm 16
1.6.3* Hensel Lifting Algorithm 16
1.6.4* Heuristic gcd 18
2 Intermezzo: Chinese Remainder Theorems 23
2.1 Chinese Remainder Theorems 24
2.2 Chinese Remainder Theorem for a Principal Ideal Domain 26
2.3 A Structure Theorem (1) 29
2.4 Nilpotents 32
2.5 Idempotents 35
2.6 A Structure Theorem (2) 39
2.7 Lagrange Formula 41
3 Cardano 47
3.1 A Tautology? 47
3.2 The Imaginary Number 48
3.3 An Impasse 51
3.4 A Tautology! 52
4 Intermezzo: Multiplicity of Roots 53
4.1 Characteristic of a Field 54
4.2 Finite Fields 55
4.3 Derivatives 57
4.4 Multiplicity 58
4.5 Separability 62
4.6 Perfect Fields 64
4.7 Squarefree Decomposition 68
5 Kronecker I: Kronecker’s Philosophy 74
5.1 Quotients of Polynomial Rings 75
5.2 The Invention of the Roots 76
5.3 Transcendental and Algebraic Field Extensions 81
5.4 Finite Algebraic Extensions 84
5.5 Splitting Fields 86
6 Intermezzo: Sylvester 91
6.1 Gauss Lemma 92
6.2 Symmetric Functions 96
6.3* Newton’s Theorem 100
6.4 The Method of Indeterminate Coefficients 106
6.5 Discriminant 108
6.6 Resultants 112
6.7 Resultants and Roots 115
7 Galois I: Finite Fields 119
7.1 Galois Fields 120
7.2 Roots of Polynomials over Finite Fields 123
7.3 Distinct Degree Factorization 125
7.4 Roots of Unity and Primitive Roots 127
7.5 Representation and Arithmetics of Finite Fields 133
7.6* Cyclotomic Polynomials 135
7.7* Cycles, Roots and Idempotents 141
7.8 Deterministic Polynomial-time Primality Test 148
8 Kronecker II: Kronecker’s Model 156
8.1 Kronecker’s Philosophy 156
8.2 Explicitly Given Fields 159
8.3 Representation and Arithmetics 164
8.3.1 Representation 164
8.3.2 Vector space arithmetics .... 165
8.3.3 Canonical representation 165
8.3.4 Multiplication 167
8.3.5 Inverse and division 167
8.3.6 Polynomial factorization 168
8.3.7 Solving polynomial equations 169
8.3.8 Monic polynomials 169
8.4 Primitive Element Theorems 170
9 Steinitz 175
9.1 Algebraic Closure 176
9.2 Algebraic Dependence and Transcendency Degree 180
9.3 The Structure of Field Extensions 184
9.4 Universal Field 186
9.5* Luroth’s Theorem 187
10 Lagrange 191
10.1 Conjugates 192
10.2 Normal Extension Fields 193
10.3 Isomorphisms 196
10.4 Splitting Fields 203
10.5 Trace and Norm 206
10.6 Discriminant 212
10.7* Normal Bases 216
11 Duval 221
11.1 Explicit Representation of Rings 221
11.2 Ring Operations in a Non-unique Representation 223
11.3 Duval Representation 224
11.4 Duval’s Model 228
12 Gauss 232
12.1 The Fundamental Theorem of Algebra 232
12.2 Cyclotomic Equations 237
13 Sturm 263
13.1* Real Closed Fields 264
13.2 Definitions 272
13.3 Sturm 275
13.4 Sturm Representation of Algebraic Reals 280
13.5 Hermite’s Method 284
13.6 Thom Codification of Algebraic Reals (1) 288
13.7 Ben-Or, Kozen and Reif Algorithm 290
13.8 Thom Codification of Algebraic Reals (2) 294
14 Galois II 297
14.1 Galois Extension 298
14.2 Galois Correspondence 300
14.3 Solvability by Radicals 305
14.4 Abel-Ruffini Theorem 314
14.5* Constructions with Ruler and Compass
318
Part two: Factorization 327
15 Prelude 329
15.1 A Computation 329
15.2 An Exercise 338
16 Kronecker III: factorization 346
16.1 Von Schubert Factorization Algorithm over the Integers 347
16.2 Factorization of Multivariate Polynomials 350
16.3 Factorization over a Simple Algebraic Extension 352
17 Berlekamp 361
17.1 Berlekamp’s Algorithm 361
17.2 The Cantor-Zassenhaus Algorithm 369
18 Zassenhaus 380
18.1 Hensel’s Lemma 381
18.2 The Zassenhaus Algorithm 389
18.3 Factorization Over a Simple Transcendental Extension 391
18.4 Cauchy Bounds 395
18.5 Factorization over the Rationals 398
18.6 Swinnerton-Dyer Polynomials 402
18.7 L3 Algorithm 405
19 Finale 415
19.1 Kronecker s Dream 415
19.2 Van der Waerden’s Example 415
Bibliography 420
Index
422
|
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| discipline | Mathematik |
| edition | 1. publ. |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T19:12:28Z |
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| isbn | 0521811546 |
| language | English |
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| physical | XIII, 423 S. |
| publishDate | 2003 |
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| publisher | Cambridge University Press |
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| series | Encyclopedia of mathematics and its applications |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | Mora, Teo Verfasser aut Solving polynomial equation systems 1 The Kronecker-Duval philosophy Teo Mora 1. publ. New York Cambridge University Press 2003 XIII, 423 S. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 88 Encyclopedia of mathematics and its applications ... (DE-604)BV016980289 1 Encyclopedia of mathematics and its applications 88 (DE-604)BV000903719 88 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010255533&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mora, Teo Solving polynomial equation systems Encyclopedia of mathematics and its applications |
| title | Solving polynomial equation systems |
| title_auth | Solving polynomial equation systems |
| title_exact_search | Solving polynomial equation systems |
| title_full | Solving polynomial equation systems 1 The Kronecker-Duval philosophy Teo Mora |
| title_fullStr | Solving polynomial equation systems 1 The Kronecker-Duval philosophy Teo Mora |
| title_full_unstemmed | Solving polynomial equation systems 1 The Kronecker-Duval philosophy Teo Mora |
| title_short | Solving polynomial equation systems |
| title_sort | solving polynomial equation systems the kronecker duval philosophy |
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