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On Construction of Saturated Distinguished Chains
Author(s) -
Singh Amrit Pal,
Khanduja Sudesh K.
Publication year - 2007
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300000218
Subject(s) - mathematics , residue field , algebraic closure , discrete valuation , chain (unit) , algebraic number , field (mathematics) , rank (graph theory) , field extension , combinatorics , discrete mathematics , pure mathematics , mathematical analysis , differential algebraic equation , ordinary differential equation , physics , astronomy , differential equation
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 K ¯ of K . For an element α ∈ K ¯ \ K , a chain α = α 0 , α 1 ,…,α r of elements of K ¯ , such that α i is of minimum degree over K with the property that ῡ(α i −1 − α i ) = sup{ῡ(α i −1 − β) | [ K (β) : K ] < [ K (α i −1 ) : K ]} and that α r ∈ 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 K ¯ ( 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.