Boston, MA: Birkhäuser. xviii, 184 p. $ 42.95; EUR 59.00; sFr. 79.00 (2002).
This book covers practically all results on index form equations for algebraic number fields obtained until now. The problem under consideration is usually the following: Given an order $O$ of an algebraic number field (usually an equation order, sometimes the maximal order) one is interested in calculating a basis for that order whose elements are successive powers of a single element, a so-called power basis. The index of the order generated by the successive powers of any element of $O$ is the value of a homogeneous multivariate polynomial of high degree, the so-called index form. Hence, the existence of a power basis is tantamount to the index form attaining the value $\pm 1$. Because of the number of variables ($n-1$ if the field degree is $n$) and the size of the degree of the index form $(n(n-1)/2)$ the task of solving index form equations is very difficult in general.
The first three chapters of this book introduce to the problem and necessary auxiliary tools like Baker’s method (yielding bounds for potential solutions), the Davenport lemma (reducing those bounds substantially) and enumeration strategies to be employed for checking the remaining potential solutions. Also, other equations like Thue and norm form equations are used in the context of solving index form equations. In chapter 4 the author discusses various aspects of index form equations if the underlying field has special properties. Then chapters 5 to 8 present what is presently known about solving index forms in fields of degree 3 to 6. The last two chapters treat the case of relative index forms. When subfields exist the relative point of view is often superior. This is demonstrated by fields of higher degree, especially of degrees 8 and 9. The final chapter 11 contains more than 20 pages of tables.
This book is a nice survey on the subject. The author has spent a considerable part of his mathematical life in studying index forms and this is in large parts a compilation of his achievements. The book is well written and is certainly to be recommended to the audience it addresses: advanced undergraduates, graduates and researchers.