On Extensions of Valuations with given Residue Field and Value Group

Let υ be a valuation on K with value group G υ , residue field k υ , rank υ = t and K ( x 1 , …, x n ) be the field of rational functions over K with n variables. If G is the direct sum of G 1 and d infinite cyclic groups where G 1 is a totally ordered group containing G υ as an ordered subgroup with [ G 1 : G υ ] < ∞ and k ′ is a finite field extension of k υ then there exists a residual transcendental extension u of υ to K ( x 1 , …, x n ) such that rank u = t + d , G u = G the algebraic closure of k υ in k υ is k ′ and trans deg k u / k υ = n − d .