Hardcover ISBN:  9780821840412 
Product Code:  SURV/124 
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MAA Member Price:  $77.40 
AMS Member Price:  $68.80 
Electronic ISBN:  9781470413514 
Product Code:  SURV/124.E 
List Price:  $81.00 
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AMS Member Price:  $64.80 

Book DetailsMathematical Surveys and MonographsVolume: 124; 2006; 288 ppMSC: 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 selfcontained. 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 valuationtheoretic 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.ReadershipGraduate students and research mathematicians interested in valuations and orderings on fields (algebra and number theory).

Table of Contents

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


Additional Material

Reviews

This book provides a thorough and wellpresented introduction to the valuations and orderings on a field, and developsa formalism of relative higher Milnor \(K\)theory of fields.
Mathematical Reviews


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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 selfcontained. 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 valuationtheoretic 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 wellpresented introduction to the valuations and orderings on a field, and developsa formalism of relative higher Milnor \(K\)theory of fields.
Mathematical Reviews