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Hilbert's Tenth Problem: Diophantine Classes and Extensions to Global Fields

Alexandra Shlapentokh
Cambridge University Press
Publication Date: 
Number of Pages: 
New Mathematical Monographs 7
[Reviewed by
Mihaela Poplicher
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In 1900, at the International Congress of Mathematicians, Hilbert presented a list of problems he considered important to be solved in the new century. The tenth problem on the list, known, of course, as Hilbert’s Tenth Problem, asked for an algorithm to determine whether a polynomial equation with integer coefficients, in several variables, has integer solutions. Many mathematicians (Martin Davis, Hilary Putnam, and Julia Robinson) worked on this problem and an answer was found in 1970 by Matiyasevich; Hilbert’s Tenth Problem is unsolvable: no such algorithm exists.

In the last thirty-five years, much work has been done to extend this result. The present book presents such extensions, to integrally closed subrings of global fields.

While this is a book containing deep results in Algebraic Number Theory, accessible in its entirety only to experienced readers, it contains some chapters that can be used in an introductory (undergraduate) course and can be used in graduate courses as well.

The book contains two Appendices: one on Recursion (computability) Theory and the other on Number Theory. These cover much of the background the reader needs. The first three chapters should be quite accessible to undergraduate students.

Overall, this is a very good book addressing a lot of readers — from undergraduate students, to graduate students and research mathematicians — and presenting important results, with a very large list of references for further reading.

Mihaela Poplicher is associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is [email protected].

 1. Introduction; 2. Diophantine classes: definition and basic facts; 3. Diophantine equivalence and diophantine decidability; 4. Integrality at finitely many primes and divisibility of order at infinitely many primes; 5. Bound equations for number fields and their consequences; 6. Units of rings of W-integers of norm 1; 7. Diophantine classes over number fields; 8. Diophantine undecidability of function fields; 9. Bounds for function fields; 10. Diophantine classes over function fields; 11. Mazur’s conjectures and their consequences; 12. Results of Poonen; 13. Beyond global fields; A. Recursion theory; B. Number theory; Bibliography; Index.