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On Construction of Saturated Distinguished Chains

Part of: Topological fields

Published online by Cambridge University Press:  21 December 2009

Amrit Pal Singh  and
Sudesh K. Khanduja
Amrit Pal Singh
Affiliation:
Department of Mathematics, Panjab University, Chandigarh-160014, India. E-mail: [email protected]
Sudesh K. Khanduja
Affiliation:
Department of Mathematics, Panjab University, Chandigarh-160014, India. E-mail: [email protected]
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Abstract

Let υ be a Henselian valuation of arbitrary rank of a field K, and let ῡ be the (unique) extension of v to a fixed algebraic closure of K. For an element α ∈ \K, a chain α = α0, α1,…,αr of elements of , such that αi is of minimum degree over K with the property that ῡ(αi−1 − αi) = sup{ῡ(αi−1 − β) | [K (β) : K] < [K (αi−1) : K]} and that αr ∈ K, is called a saturated distinguished chain for α with respect to (K, υ). The notion of a saturated distinguished chain has been used to obtain results about the irreducible polynomials over any complete discrete rank one valued field K and to determine various arithmetic and metric invariants associated to elements of (cf. [J. Number Theory, 52 (1995), 98–118.] and [J. Algebra, 266 (2003), 14–26]). In this paper, a method is described of constructing a saturated distinguished chain for α, and also determining explicitly some invariants associated to α, when the degree of the extension K (α)/K is not divisible by the characteristic of the residue field of υ.

MSC classification

Secondary: 12J10: Valued fields 12J25: Non-Archimedean valued fields
Type
Research Article
Information
Mathematika , Volume 54 , Issue 1-2 , December 2007 , pp. 59 - 65
Copyright
Copyright © University College London 2007

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References

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