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Prolongations of valuations to simple transcendental extensions with given residue field and value group
Author(s) -
Khanduja Sudesh K.
Publication year - 1991
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300006732
Subject(s) - mathematics , residue field , abelian group , transcendental number , corollary , algebraic number , residue (chemistry) , pure mathematics , algebraic extension , valuation (finance) , combinatorics , discrete mathematics , field (mathematics) , mathematical analysis , differential equation , ordinary differential equation , chemistry , finance , economics , biochemistry , differential algebraic equation
Let K 0 (x) be a simple transcendental extension of a field K 0 , υ 0 be a valuation of K 0 with value group G 0 and residue field K 0 . SupposeG 0 ⊆ G 1 ⊆ G is an inclusion of totally ordered abelian groups with [ G 1 : G 0 ] < ∞ such that G is the direct sum of G 1 and an infinite cyclic group. It is proved that there exists an (explicitly constructible) valuation υ of K 0 ( x ) extending υ 0 such that the value group of υ is G and its residue field is k , where k is a given finite extension of k 0 . This is analogous to a result of Matignon and Ohm [2, Corollary 3.2] for residually non‐algebraic prolongations of υ 0 to K 0 ( x ).