Let (K, v) be a valued field. We reinterpret some results of MacLane and Vaquié on
extensions of v to valuations on the polynomial ring K [x]. We introduce certain MacLane …

Let (K, v) be a discrete valued field and let x be an indeterminate. In 1936, MacLane
determined all valuations on K (x) extending v. His work has been reviewed and generalized …

We associate to any given finite set of valuations on the polynomial ring in two variables
over an algebraically closed field a numerical invariant whose positivity characterizes the …

Let V be a valuation domain with quotient field K. We show how to describe all extensions of
V to K (X) when the V-adic completion K ˆ is algebraically closed, generalizing a similar …

For a fixed irreducible polynomial F we study the set VF of all valuations on K [x] bounded by
valuations whose support is (F). The first main result presents a characterization for …

Given a valuation v on a field K, an extension v¯ to an algebraic closure K¯ and an
extension w to K (X). In this paper, we study common extensions w¯ of both w and v¯ to the …

We want to determine all the extensions of a valuation $\nu $ of a field $ K $ to a cyclic
extension $ L $ of $ K $, ie $ L= K (x) $ is the field of rational functions of $ x $ or $ L= K …

Let {R,,,} be a system of topological rings and Qa open subrings of Ra. We consider the set
R of all vectors a=(aa), where aa are elements in Ra and which belong to Qa, except for a …

Assume that $(L, v) $ is a finite Galois extension of a valued field $(K, v) $. We give an
explicit construction of the valuation ring $\cO_L $ of $ L $ as an $\cO_K $-algebra, and an …

Let $(K,\nu) $ be a valued field and $ K (x) $ a simple purely transcendental extension of $ K
$. In the nineteen thirties, in order to study the possible extensions of $\nu $ to $ K (x) $, S …