Valuations, Orderings, and Milnor K-Theory

Hardcover (Print)
Buy New
Buy New from
Used and New from Other Sellers
Used and New from Other Sellers
from $27.00
Usually ships in 1-2 business days
(Save 66%)
Other sellers (Hardcover)
  • All (8) from $27.00   
  • New (6) from $70.99   
  • Used (2) from $27.00   


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.

Read More Show Less

Product Details

  • ISBN-13: 9780821840412
  • Publisher: American Mathematical Society
  • Publication date: 2/28/2006
  • Series: Mathematical Surveys and Monographs, #124
  • Pages: 288
  • Product dimensions: 7.20 (w) x 10.30 (h) x 0.90 (d)

Table of Contents

Ch. 1 Preliminaries on Abelian groups 5
Ch. 2 Ordered Abelian groups 15
Ch. 3 Valuations 37
Ch. 4 Examples of valuations 47
Ch. 5 Coarsenings of valuations 55
Ch. 6 Orderings 63
Ch. 7 The tree of localities 69
Ch. 8 Topologies 75
Ch. 9 Complete fields 81
Ch. 10 Approximation theorems 87
Ch. 11 Canonical valuations 95
Ch. 12 Valuations of mixed characteristics 107
Ch. 13 Infinite Galois theory 125
Ch. 14 Valuations in field extensions 127
Ch. 15 Decomposition groups 133
Ch. 16 Ramification theory 141
Ch. 17 The fundamental equality 151
Ch. 18 Hensel's lemma 161
Ch. 19 Real closures 175
Ch. 20 Coarsening in algebraic extensions 187
Ch. 21 Intersections of decomposition groups 193
Ch. 22 Sections 199
Ch. 23 K-structures 209
Ch. 24 Milnor K-rings of fields 217
Ch. 25 Milnor K-rings and orderings 225
Ch. 26 K-rings and valuations 231
Ch. 27 K-rings of wild valued fields 247
Ch. 28 Decompositions of K-rings 253
Ch. 29 Realization of k-structures 263
Read More Show Less

Customer Reviews

Be the first to write a review
( 0 )
Rating Distribution

5 Star


4 Star


3 Star


2 Star


1 Star


Your Rating:

Your Name: Create a Pen Name or

Barnes & Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at or to post customer reviews. Please see our Terms of Use for more details.

What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

Reviews should not contain any of the following:

  • - HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
  • - Time-sensitive information such as tour dates, signings, lectures, etc.
  • - Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
  • - Comments focusing on the author or that may ruin the ending for others
  • - Phone numbers, addresses, URLs
  • - Pricing and availability information or alternative ordering information
  • - Advertisements or commercial solicitation


  • - By submitting a review, you grant to Barnes & and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Terms of Use.
  • - Barnes & reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & also reserves the right to remove any review at any time without notice.
  • - See Terms of Use for other conditions and disclaimers.
Search for Products You'd Like to Recommend

Recommend other products that relate to your review. Just search for them below and share!

Create a Pen Name

Your Pen Name is your unique identity on It will appear on the reviews you write and other website activities. Your Pen Name cannot be edited, changed or deleted once submitted.

Your Pen Name can be any combination of alphanumeric characters (plus - and _), and must be at least two characters long.

Continue Anonymously

    If you find inappropriate content, please report it to Barnes & Noble
    Why is this product inappropriate?
    Comments (optional)