Maximally Complete Fields
نویسنده
چکیده
Kaplansky proved in 1942 that among ail 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 Qor Fp , or the /?-adic valuation on Q), there is one that is maximal in the strong sensé 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 resuit. The field turns out to be a MaFcev-Neumann ring or a /?-adic version of a MaFcev-Neumann ring in which the éléments are formai séries of the form Y, ge s a§P s wnere S is a well-ordered subset of G and the a/ s are residue class représentatives. We conclude with some remarks on the p-adic MaPcev-Neumann field containing Q^.
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