This book is an updated version of the author’s book published in 2002 [Boston, MA: Birkhäuser (2002; Zbl 1016.11059)]. The first edition attracted reasonable interest among the researchers. The topic of power integral bases and index-form equations is an active research topic, and the author is the world-leading expert in that subject with many collaborators and PhD students, thus there was a natural need for an updated version of the book. The book covers and systematizes many papers by the author and his collaborators published in respectable international mathematical journals.
After describing the basic concepts and summarizing some important results on power integral bases in Chapter 1, in Chapter 2, the author collects the main tools for solving Diophantine equations studied in the book: Baker’s method of linear forms in logarithms, reduction methods and enumeration algorithms. In Chapters 3–6, these tools are applied to classical Thue equations, inhomogeneous Thue equations, relative Thue equations and norm form equations. After a general overview of the structure of index forms in Chapter 7, in Chapters 8–11 are described algorithms that can be used for solving index form equations in cubic, quartic, quintic and sextic number fields. Chapter 12 is devoted to pure fields, with a lot of interesting properties. In Chapters 13 and 14, the problem of relative power integral bases in cubic and quartic relative extensions of fields is considered, while in Chapter 15, the author considers power integral bases in some special types of higher degree number fields. Finally, in Chapter 16, the results of computations are provided: tables of solutions of Diophantine equations and tables of generators of power integral bases in cubic, quartic and sextic fields.
The book is recommended to PhD students and researchers in the field of Diophantine equations. It can be used as a textbook for a specialized graduate course in Thue and index-form equations and as an additional reading for a general course in Diophantine equations.