MAXIMALLY COMPLETE FIELDS

Kaplansky proved in 1942 that among all fields with a valuation having a given divisible value group G, a given algebraically closed residue field R, and a given restriction to the minimal subfield (either the trivial valuation on Q or Fp, or the p-adic valuation on Q), there is one that is maximal in the strong sense that every other can be embedded in it. In this paper, we construct this field explicitly and use the explicit form to give a new proof of Kaplansky’s result. The field turns out to be a Mal’cev-Neumann ring or a p-adic version of a Mal’cev-Neumann ring in which the elements are formal series of the form ∑ g∈S αgp g where S is a well-ordered subset of G and the αg ’s are residue class representatives. We conclude with some remarks on the p-adic Mal’cev-Neumann field containing Q̄p.