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We give a characterization of finitely ramified $\omega$-pseudo completevalued fields of mixed characteristic $(0, p)$, with fixed residue field $k$and value group $G$ of cardinality $\aleph_{1}$, in terms of a Hahn-likeconstruction over the Cohen field $C(k)$, modulo the Continuum Hypothesis. Thisis a generalization of results due to Ax and Kochen in '65 for formally$p$-adic fields and by Kochen in '74 for unramified valued fields with perfectresidue field. We consider a more general context of finitely ramified valuedfields of mixed characteristic with arbitrary residue field and across-section.

A characterization of pseudo complete finitely ramified valued fields through a Hahn-like construction