It is known that if L is a
separable, finite dimensional extension of a field K and if v is a proper valuation
(absolute value) on K, then each ring topology on L whose restriction to K is
the topology 𝒯v defined on K by v is the supremum of a finite family of
valuation (absolute valued) topologies. We give a characterization of the fields
K and L and the valuations (absolute values) v on K for which each ring
topology on L extending 𝒯v is the supremum of a family of valuation (absolute
valued) topologies on K when L is an arbitrary finite dimensional extension of
K.
