An error was encountered while trying to add the item to the cart. Please try again.
Copy To Clipboard
Successfully Copied!
Valuations, Orderings, and Milnor $K$-Theory

Ido Efrat Ben-Gurion University of the Negev, Be’er-Sheva, Israel
Available Formats:
Hardcover ISBN: 978-0-8218-4041-2
Product Code: SURV/124
List Price: $86.00 MAA Member Price:$77.40
AMS Member Price: $68.80 Electronic ISBN: 978-1-4704-1351-4 Product Code: SURV/124.E List Price:$81.00
MAA Member Price: $72.90 AMS Member Price:$64.80
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $129.00 MAA Member Price:$116.10
AMS Member Price: $103.20 Click above image for expanded view Valuations, Orderings, and Milnor$K$-Theory Ido Efrat Ben-Gurion University of the Negev, Be’er-Sheva, Israel Available Formats:  Hardcover ISBN: 978-0-8218-4041-2 Product Code: SURV/124  List Price:$86.00 MAA Member Price: $77.40 AMS Member Price:$68.80
 Electronic ISBN: 978-1-4704-1351-4 Product Code: SURV/124.E
 List Price: $81.00 MAA Member Price:$72.90 AMS Member Price: $64.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$129.00 MAA Member Price: $116.10 AMS Member Price:$103.20
• Book Details

Mathematical Surveys and Monographs
Volume: 1242006; 288 pp
MSC: Primary 12; Secondary 19;

This monograph is a comprehensive exposition of the modern theory of valued and ordered fields. It presents the classical aspects of such fields: their arithmetic, topology, and Galois theory. Deeper cohomological aspects are studied in its last part in an elementary manner. This is done by means of the newly developed theory of generalized Milnor $K$-rings. The book emphasizes the close connections and interplay between valuations and orderings, and to a large extent, studies them in a unified manner.

The presentation is almost entirely self-contained. In particular, the text develops the needed machinery of ordered abelian groups. This is then used throughout the text to replace the more classical techniques of commutative algebra. Likewise, the book provides an introduction to the Milnor $K$-theory.

The reader is introduced to the valuation-theoretic techniques as used in modern Galois theory, especially in applications to birational anabelian geometry, where one needs to detect valuations from their “cohomological footprints”. These powerful techniques are presented here for the first time in a unified and elementary way.

Graduate students and research mathematicians interested in valuations and orderings on fields (algebra and number theory).

• Chapters
• 1. Preliminaries on Abelian groups
• 2. Ordered Abelian groups
• 3. Valuations
• 4. Examples of valuations
• 5. Coarsenings of valuations
• 6. Orderings
• 7. The tree of localities
• 8. Topologies
• 9. Complete fields
• 10. Approximation theorems
• 11. Canonical valuations
• 12. Valuations of mixed characteristics
• 13. Infinite Galois theory
• 14. Valuations in field extensions
• 15. Decomposition groups
• 16. Ramification theory
• 17. The fundamental equality
• 18. Hensel’s lemma
• 19. Real closures
• 20. Coarsening in algebraic extensions
• 21. Intersections of decomposition groups
• 22. Sections
• 23. $\kappa$-structures
• 24. Milnor $K$-rings of fields
• 25. Milnor $K$-rings and orderings
• 26. $K$-rings and valuations
• 27. $K$-rings of wild valued fields
• 28. Decompositions of $K$-rings
• 29. Realization of $\kappa$-structures

• Reviews

• This book provides a thorough and well-presented introduction to the valuations and orderings on a field, and developsa formalism of relative higher Milnor $K$-theory of fields.

Mathematical Reviews
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1242006; 288 pp
MSC: Primary 12; Secondary 19;

This monograph is a comprehensive exposition of the modern theory of valued and ordered fields. It presents the classical aspects of such fields: their arithmetic, topology, and Galois theory. Deeper cohomological aspects are studied in its last part in an elementary manner. This is done by means of the newly developed theory of generalized Milnor $K$-rings. The book emphasizes the close connections and interplay between valuations and orderings, and to a large extent, studies them in a unified manner.

The presentation is almost entirely self-contained. In particular, the text develops the needed machinery of ordered abelian groups. This is then used throughout the text to replace the more classical techniques of commutative algebra. Likewise, the book provides an introduction to the Milnor $K$-theory.

The reader is introduced to the valuation-theoretic techniques as used in modern Galois theory, especially in applications to birational anabelian geometry, where one needs to detect valuations from their “cohomological footprints”. These powerful techniques are presented here for the first time in a unified and elementary way.

Graduate students and research mathematicians interested in valuations and orderings on fields (algebra and number theory).

• Chapters
• 1. Preliminaries on Abelian groups
• 2. Ordered Abelian groups
• 3. Valuations
• 4. Examples of valuations
• 5. Coarsenings of valuations
• 6. Orderings
• 7. The tree of localities
• 8. Topologies
• 9. Complete fields
• 10. Approximation theorems
• 11. Canonical valuations
• 12. Valuations of mixed characteristics
• 13. Infinite Galois theory
• 14. Valuations in field extensions
• 15. Decomposition groups
• 16. Ramification theory
• 17. The fundamental equality
• 18. Hensel’s lemma
• 19. Real closures
• 20. Coarsening in algebraic extensions
• 21. Intersections of decomposition groups
• 22. Sections
• 23. $\kappa$-structures
• 24. Milnor $K$-rings of fields
• 25. Milnor $K$-rings and orderings
• 26. $K$-rings and valuations
• 27. $K$-rings of wild valued fields
• 28. Decompositions of $K$-rings
• 29. Realization of $\kappa$-structures
• This book provides a thorough and well-presented introduction to the valuations and orderings on a field, and developsa formalism of relative higher Milnor $K$-theory of fields.

Mathematical Reviews
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.