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**Advanced topics in computational number theory.**
*(English)*
Zbl 0977.11056

Graduate Texts in Mathematics. 193. New York, NY: Springer. xv, 578 p. (2000).

The first five chapters of this work accomplish the main purpose, to form an independent computational treatment of class field theory stressing the Hilbert class field. This treatment complements the author’s earlier work [A Course in Computational Algebraic Number Theory, Springer Graduate Texts 138 (1996; Zbl 0786.11071)].

With an algebraic (module-theoretic) introduction in Chapter 1 (and appendix), relative fields are explored in Chapter 2 and the basic (Takagi) correspondence of Abelian extensions to ray ideal groups in Chapter 3. (This is reinforced computationally in Chapter 4). Chapter 5 develops Kummer extensions as the building block of class fields. These chapters, rich in examples, would provide a suitable textbook for students advanced enough to absorb the initial impact of theorems in full generality. Later on, Chapters 8 and 9 deal with the cubic case and the construction of tables of simplest cases (of small unit rank). The tables are arranged not only by discriminant but also by class number (i.e., by Hilbert class field). The author shows considerable self-restraint in not saturating the reader with titanic numerical illustrations. The role of ideals becoming principal in an extension field is given priority over the role of the splitting of prime ideals, but fortunately these concepts merge for the Hilbert class field.

Chapter 6 supplements the examples with a sprinkling of historical and recent constructions such as complex multiplication, Stark’s conjecture, Shimura’s reciprocity, Dedekind sums, etc. Chapter 7 brings in relative units and also the dubious concept of relative class number. (Do we inject ideals in the larger field or take norms into the smaller?) Relative norm algorithms are treated in greater detail as needed for the chain of Kummer extensions. For informed students there are topics for several different seminars. The choice of material here is bound to be more subjective than in the classical theory. For example, the study of norm equations would take on a different aspect (exciting to some but not to others) if norm equations were treated as Diophantine equations instead of ideal factorizations [see N. Smart, The Algorithmic Resolution of Diophantine Equations, Cambridge (1998; Zbl 0907.11001)].

Overall, this book is highly algorithmic. Indeed, there is a useful appendix on computer algebra systems and web sites. This book is in many ways an indispensible tool for algebraic number theorists.

With an algebraic (module-theoretic) introduction in Chapter 1 (and appendix), relative fields are explored in Chapter 2 and the basic (Takagi) correspondence of Abelian extensions to ray ideal groups in Chapter 3. (This is reinforced computationally in Chapter 4). Chapter 5 develops Kummer extensions as the building block of class fields. These chapters, rich in examples, would provide a suitable textbook for students advanced enough to absorb the initial impact of theorems in full generality. Later on, Chapters 8 and 9 deal with the cubic case and the construction of tables of simplest cases (of small unit rank). The tables are arranged not only by discriminant but also by class number (i.e., by Hilbert class field). The author shows considerable self-restraint in not saturating the reader with titanic numerical illustrations. The role of ideals becoming principal in an extension field is given priority over the role of the splitting of prime ideals, but fortunately these concepts merge for the Hilbert class field.

Chapter 6 supplements the examples with a sprinkling of historical and recent constructions such as complex multiplication, Stark’s conjecture, Shimura’s reciprocity, Dedekind sums, etc. Chapter 7 brings in relative units and also the dubious concept of relative class number. (Do we inject ideals in the larger field or take norms into the smaller?) Relative norm algorithms are treated in greater detail as needed for the chain of Kummer extensions. For informed students there are topics for several different seminars. The choice of material here is bound to be more subjective than in the classical theory. For example, the study of norm equations would take on a different aspect (exciting to some but not to others) if norm equations were treated as Diophantine equations instead of ideal factorizations [see N. Smart, The Algorithmic Resolution of Diophantine Equations, Cambridge (1998; Zbl 0907.11001)].

Overall, this book is highly algorithmic. Indeed, there is a useful appendix on computer algebra systems and web sites. This book is in many ways an indispensible tool for algebraic number theorists.

Reviewer: Harvey Cohn (San Diego)