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This monograph investigates algorithms for determining power integral bases in algebraic number fields. The problem has classical roots and leads to the problem of solving the corresponding index form equations that are often reduced to more classical equations, such as various types of Thue equations. §The reader is introduced to the best-known methods for solving several types of diophantine equations using Baker-type estimates, reduction methods, and enumeration algorithms. These methods can be useful for other types of diophantine equations not included in the book. §Several interesting properties of number fields are examined. Some infinite parametric families of fields are also considered as well as the resolution of the corresponding infinite parametric families of diophantine equations. The text is illustrated with several tables of various number fields, including their data on power integral bases. §Advanced undergraduates and graduates will benefit from this exposition of methods for solving some classical types of diophantine equations. Researchers in the field will find new applications for the tools presented throughout the book.
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