Abstract
AbstractWe give conditions for Kaplansky fields to admit infinite towers of Galois defect extensions of prime degree. As proofs of the presented facts are constructive, this provides examples of constructions of infinite towers of Galois defect extensions of prime degree. We also give a constructive proof of the fact that a henselian Kaplansky field cannot be defectless-by-finite.
Highlights
IntroductionWe denote by (K, v) a field K equipped with a valuation v
In this paper, we denote by (K, v) a field K equipped with a valuation v
We show that if a Kaplansky field of positive characteristic p admits a Galois defect extension it admits an infinite tower of such extensions
Summary
We denote by (K, v) a field K equipped with a valuation v. See Theorem 1 of [16], which shows the equivalence of conditions (K1) - (K3) with the original “hypothesis A” assumed by Kaplansky He gave an example of a valued field admitting nonisomorphic maximal immediate extensions (cf [6, Section 5]). In the present paper we give conditions for Kaplansky fields to admit infinite towers of Galois defect extensions of prime degree. We show that if a Kaplansky field of positive characteristic p admits a Galois defect extension it admits an infinite tower of such extensions (cf Theorem 3.2). We give conditions for a valued field of characteristic 0 and residue characteristic 2 to admit a infinite tower of defect extensions of degree 2 (cf Theorem 3.6, Corollary 3.7 and Corollary 3.8). The facts mentioned above allow us to give a constructive alternative proof of the fact that Kaplansky fields cannot be defectless-by-finite (cf. Corollary 3.10)
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