Description
Book SynopsisPresents a comprehensive exposition of the modern theory of valued and ordered fields. This title presents the classical aspects of such fields: their arithmetic, topology, and Galois theory. It emphasizes the close connections and interplay between valuations and orderings, and to a large extent, studies them in a unified manner.
Table of ContentsPart I. Abelian Groups: Preliminaries on abelian groups Ordered abelian groups Part II. Valuations and orderings: Valuations Examples of valuations Coarsenings of valuations Orderings The tree of localities Topologies Complete fields Approximation theorems Canonical valuations Valuations of mixed characteristics Part III. Galois Theory: Infinite Galois theory Valuations in field extensions Decomposition groups Ramification theory The fundamental equality Hensel's lemma Real closures Coarsening in algebraic extensions Intersections of decompositions groups Sections Part IV. $K$-rings: $\kappa$-structures Milnor $K$-rings of fields Milnor $K$-rings and orderings $K$-rings and valuations $K$-rings of wild valued fields Decomposition of $K$-rings Realization of $\kappa$-structures Bibliography Glossary of notation Index.
