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 properties of number fields and new applications. The text is illustrated with several tables of various number fields including their data on power integral bases. Good resource for solving classical types of diophantine equations. Aimed at advanced undergraduate graduate students and researchers. 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.