INTRODUCTION xi

Part III of the book. However, by their mere definitions, valuations and orderings

are primarily related to the multiplicative group Fx of the field F , and much can be

said when studying them in this context. This approach has become dominant in

the ordered field case (as in [Lam2]). However, it is our feeling that in the valuation

case this viewpoint has been somewhat neglected in favor of the Galois-theoretic

one. Therefore, in addition to presenting the classical theory of Galois groups of

valued field extensions, we devote several sections (in Parts II and IV of the book)

to developing the theory with emphasis on subgroups S of Fx. In particular, we

focus on valuations satisfying a natural condition called S-compatibility, which is

the analog of Henselity in the multiplicative group context.

Part IV takes this approach one step further, and studies the Milnor K-theory

of valued and ordered fields F. We recall that the Milnor K-group of F of degree

r is just the tensor product F x (g)^ • • • Sz Fx (r times) modulo the simple relations

ai g) • • • (g) a

r

= 0 whenever a^ + a j = 1 for some i j . Several important results

(or conjectures) in arithmetic geometry indicate that there should be some kind of

parallelism between Milnor's K-theory and Galois theory of fields. For instance,

the Bloch-Kato conjecture predicts a canonical isomorphism between K^1 (F)/n and

the Galois cohomology group Hr(F, /if r ) (where r 0 and n 1 are integers with

c h a r F / n , and the cohomology is with respect to the r-times twisted cyclotomic

action); this has been proved in several important cases by A.S. Merkurjev, A.A.

Suslin, M. Rost, V. Voevodsky, and others (see §24.3). It is therefore not surprising

that large parts of the Galois theory of valued and ordered fields have analogs

in this natural framework of Milnor's K-theory. These analogs will be presented

in Part IV. In some sense, this shift of viewpoint resembles the introduction of

the iC-theoretic approach to higher class field theory, complementing the earlier

Galois-theoretic approach (see [FV, Appendix B] and [FK]).

Finally, there has been much interest lately in construction of non-trivial val-

uations on fields. Such constructions emerged in the context of ordered fields (in

particular, L. Brocker's "trivialization of fans" theorem [Brl]), and later in an

elementary and explicit way by B. Jacob, R. Ware, J.K. Arason, R. Elman, and

Y.S. Hwang ([Jl], [War2], [AEJ], [HwJ]). Such constructions became especially

important in recent years in connection with the so-called birational anabelian ge-

ometry. This line of research originated from ideas of A. Grothendieck ([Gl], [G2])

as well as from works of J. Neukirch ([Nl], [N2]). Here one wants to recover the

arithmetic structure of a field (if possible, up to an isomorphism) from its vari-

ous canonical Galois groups. The point is that usually the first step is to recover

enough valuations from their cohomological (or K-theoretic) "footprints1'; see, e.g.,

[BoT], [Efl], [Ef7], [EfF], [NSW , Ch. XII], [PI], [P2], [P3], [Sp], [Sz] for more

details. In §11 we give a new presentation of the above-mentioned line of elemen-

tary constructions, based on the coarsening relation among valuations. While these

constructions were considered for some time to be somewhat mysterious, they fit

very naturally into the multiplicative group approach as discussed above, especially

when one uses the K-theoretic language. In §26 we use this language to prove the

main criterion for the existence of "optimal" valuations, as is required in the appli-

cations to the birational anabelian geometry. This is further related to the notion

of fans in the theory of ordered fields, thus closing this fruitful circle of ideas that

began with [Brl].

The prerequisites of this book are quite minimal. We assume a good algebraic

knowledge at a beginning graduate level, including of course familiarity with general