A field equipped with a valuation and satisfying Kaplansky's hypothesis A, as introduced in [a2]. Let denote the characteristic of the residue field if it is a positive prime; otherwise, set . Hypothesis A requires that:
i) the value group is a -divisible group;
ii) for every additive polynomial (cf. [a5]) with coefficients in and every , the equation has a solution in . Requirement ii) implies that is a perfect field. Using Galois cohomology, G. Whaples [a6] showed that ii) means that the degree of every finite extension of is prime to . An elementary proof of this was given by F. Delon in 1981.
I. Kaplansky considered immediate extensions (cf. Valuation) of fields with valuations. He used pseudo-Cauchy sequences (also called Ostrowski nets), which were introduced by A. Ostrowski in 1935. These are analogues of Cauchy sequences (cf. Cauchy sequence) for the metric induced by the valuation, but their limits need not be unique; for this reason, they are called pseudo-limits.
In [a3], W. Krull proved that there always exist maximal immediate extensions (in order to apply the Zorn lemma, Krull gave an upper bound for the cardinality of immediate extensions of a fixed field; an elegant deduction of this bound was later given by K.A.H. Gravett in [a1]). Kaplansky showed that if satisfies hypothesis A, then its maximal immediate extensions are unique up to a valuation-preserving isomorphism over . The same holds for the maximal immediate algebraic extensions, and this fact can also be shown via a Galois-theoretic interpretation of hypothesis A (cf. [a4]). The former result follows from the latter by a theorem of Kaplansky (which has a certain analogue in the theory of real closed fields, cf. Real closed field): If has no non-trivial immediate algebraic extensions, then the isomorphism type of an immediate extension over is determined by the pseudo-Cauchy sequences in that have pseudo-limit .
See also Model theory of valued fields.
|[a1]||K.A.H. Gravett, "Note on a result of Krull" Proc. Cambridge Philos. Soc. , 52 (1956) pp. 379|
|[a2]||I. Kaplansky, "Maximal fields with valuations I" Duke Math. J. , 9 (1942) pp. 303–321|
|[a3]||W. Krull, "Allgemeine Bewertungstheorie" J. Reine Angew. Math. , 167 (1932) pp. 160–196|
|[a4]||F.-V. Kuhlmann, M. Pank, P. Roquette, "Immediate and purely wild extensions of valued fields" Manuscr. Math. , 55 (1986) pp. 39–67|
|[a5]||S. Lang, "Algebra" , Addison-Wesley (1974)|
|[a6]||G. Whaples, "Galois cohomology of additive polynomials and -th power mappings of fields" Duke Math. J. , 24 (1957) pp. 143–150|
Kaplansky field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kaplansky_field&oldid=14681