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.
Part 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.