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On extensions generated by roots of lifting polynomials
Author(s) -
Bhatia Saurabh,
Khanduja Sudesh K.
Publication year - 2002
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/s0025579300016107
Subject(s) - monic polynomial , mathematics , residue field , algebraic closure , combinatorics , lift (data mining) , field (mathematics) , pure mathematics , discrete mathematics , polynomial , mathematical analysis , ordinary differential equation , differential equation , differential algebraic equation , computer science , data mining
Let v be a Henselian valuation of any rank of a field K andv ¯its unique prolongation to a fixed algebraic closureK ¯of K having value groupG ¯ . For any subfield L ofK ¯ , let R ( L ) denote the residue field of the valuation obtained by restrictingv ¯to L . Using the canonical homomorphism from the valuation ring of v onto its residue field R ( K ), one can lift any monic irreducible polynomial with coefficients in R ( K ) to yield a monic irreducible polynomial with coefficients in K . In an attempt to generalize this concept, Popescu and Zaharescu introduced the notion of lifting with respect to a ( K, v )‐minimal pair ( α, δ ) belonging toK ¯×G ¯ . As in the case of usual lifting, a given monic irreducible polynomial Q ( y ) belonging to R ( K ( α ))[ y ] gives rise to several monic irreducible polynomials over K which are obtained by lifting with respect to a fixed ( K, v )‐minimal pair (α, δ). If F , F 1 are two such lifted polynomials with coefficients in K having roots θ, θ 1 , respectively, then it is proved in the present paper thatv ¯ ( K ( θ ) ) = v ¯ ( K ( θ 1 ) ) , R ( K ( θ 1 ) ) ; in case ( K, v ) is a tame field, it is shown that K ( θ ) and K ( θ 1 ) are indeed K ‐isomorphic.