Abstract
Let A1,...,Ak be pairwise independent valuation rings of K. Prescribing extensions Δij. of the value group Γj and extensions\(\mathfrak{L}_i^j\) of the residue field\(H^j\) of Aj (i=1,...,rj) such that\(\sum\limits_{i = 1}^{r^j } {(\Delta _i^j :\Gamma ^j )} \cdot [\mathfrak{L}_i^j :H^j ] = n\), we provide sufficient conditions for the existence of a separable field extension L of K of degree n with exactly rj pairwise independent valuation rings Bij lying over Aj, which have Δij as value groups and\(\mathfrak{L}_i^j\) as residue fields.
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