Generalized Hensel ' S Lemma

  • SUDESH K. KHANDUJA
  • Published 2008

Abstract

Let (X, u) be a complete, rank-1 valued field with valuation ring Ra and residue field fc0. Let \f be the Gaussian extension of the valuation v to a simple transcendental extension K(x) defined by v"(J2ii') = minjMa,)}. The classical Hensel's lemma asserts that if polynomials F(x), G0(x), H0(x) in R,[x) are such that (i) v'(F(x) G0(x)//0(x)) > 0, (ii) the leading coefficient of G0(x) has K-valuation zero, (iii) there are polynomials A(x), B{x) belonging to the valuation ring of v* satisfying v*{A(X)G0(X) + B(x)//0(x) 1) > 0, then there exist G(x), H(x) in K[x] such that (a) F(x) = G(x)ff(x), (b) deg G(x) = deg G0(x), (c) »*(G(x)GoM) > 0, v"(H(x) — H0(x)) > 0. In this paper, our goal is to prove an analogous result when v x is replaced by any prolongation w of v to K(x), with the residue field of w a transcendental extension of k0.

Cite this paper

@inproceedings{KHANDUJA2008GeneralizedH, title={Generalized Hensel ' S Lemma}, author={SUDESH K. KHANDUJA}, year={2008} }