Some Examples of Valued Fields

The purpose of this paper is to use generalized power series to provide two classes of examples of valued elds. In the rst set of examples the valued elds are henselian, in general not maximal, with given residue eld and value group. In the second set of examples, the valued elds are all maximal and their families of residue elds and value groups form chains. In both cases, these elds possess additional interesting properties. To begin, I shall indicate the construction of the generalized power series elds (see Ri ], El & Ri]). Then, I shall recall the results obtained in Ri 2], which concern the lifting of algebraic properties from the residue eld and the value group to the eld of power series. 1. Let R be a commutative eld, let S be a torsion-free abelian group. Assume that S is endowed with a compatible partial order , that is, if s; t; u 2 S and s t, then s + u t + u. Let A be the set of all mappings f : S ! R, whose support supp(f) = fs 2 Sjf(s) 6 = 0g does not contain any innnite descending chain (the support is artinian), nor any innnite subset of pairwise order-incomparable elements (the support is narrow). With pointwise addition, A is an abelian group. Let s 2 S and f; g 2 A, then f(t; u) 2 S Sjt + u = s; f(t) 6 = 0; g(u) 6 = 0g is nite. This allows to deene the product fg as follows: (fg)(s) = X t+u=s f(t)g(u) for every s 2 S