Diophantine Equations and Power Integral Bases New Computational Methods
This monograph investigates algorithms for determining power integral bases in algebraic number fields. It introduces the best-known methods for solving several types of diophantine equations using Baker-type estimates, reduction methods, and enumeration algorithms. Particular emphasis is placed on...
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| Format: | eBook |
| Language: | English |
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Boston, MA
Birkhäuser Boston
2002, 2002
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| Edition: | 1st ed. 2002 |
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| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 1 Introduction
- 1.1 Basic concepts
- 1.2 Related results
- 2 Auxiliary Results, Tools
- 2.1 Baker’s method, effective finiteness theorems
- 2.2 Reduction
- 2.3 Enumeration methods
- 2.4 Software, hardware
- 3 Auxiliary Equations
- 3.1 Thue equations
- 3.2 Inhomogeneous Thue equations
- 3.3 Relative Thue equations
- 3.4 The resolution of norm form equations
- 4 Index Form Equations in General
- 4.1 The structure of the index form
- 4.2 Using resolvents
- 4.3 Factorizing the index form when proper subfields exist
- 4.4 Composite fields
- 5 Cubic Fields
- 5.1 Arbitrary cubic fields
- 5.2 Simplest cubic fields
- 6 Quartic Fields
- 6.1 Algorithm for arbitrary quartic fields
- 6.2 Simplest quartic fields
- 6.3 An interesting application to mixed dihedral quartic fields
- 6.4 Totally complex quartic fields
- 6.5 Bicyclic biquadratic number fields
- 7 Quintic Fields
- 7.1 Algorithm for arbitrary quintic fields
- 7.2 Lehmer’s quintics
- 8 Sextic Fields
- 8.1 Sextic fields with a quadratic subfield
- 8.2 Sextic fields with a cubic subfield
- 8.3 Sextic fields as composite fields
- 9 Relative Power Integral Bases
- 9.1 Basic concepts
- 9.2 Relative cubic extensions
- 9.3 Relative quartic extensions
- 10 Some Higher Degree Fields
- 10.1 Octic fields with a quadratic subfield
- 10.2 Nonic fields with cubic subfields
- 10.3 Some more fields of higher degree
- 11 Tables
- 11.1 Cubic fields
- 11.2 Quartic fields
- 11.3 Sextic fields
- References
- Author Index