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Valuations, Orderings, and Milnor $K$-Theory

About this Title

Ido Efrat, Ben-Gurion University of the Negev, Be’er-Sheva, Israel

Publication: Mathematical Surveys and Monographs
Publication Year: 2006; Volume 124
ISBNs: 978-0-8218-4041-2 (print); 978-1-4704-1351-4 (online)
DOI: https://doi.org/10.1090/surv/124
MathSciNet review: MR2215492
MSC: Primary 12J10; Secondary 11R70, 12J15, 19D45

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Table of Contents

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Front/Back Matter

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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