Khanduja, Sudesh K. ; Saha, Jayanti
(1999)
*Generalized Hensel's lemma*
Proceedings of the Edinburgh Mathematical Society, 42
(3).
pp. 469-480.
ISSN 0013-0915

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Official URL: http://journals.cambridge.org/action/displayAbstra...

Related URL: http://dx.doi.org/10.1017/S0013091500020460

## Abstract

Let (K, v) be a complete, rank-1 valued field with valuation ring R_{v}, and residue field k_{v}. Let v^{x} be the Gaussian extension of the valuation v to a simple transcendental extension K(x) defined by v^{x}(∑,a_{1}x^{3}=min_{i}{v(a_{i})}inline1 The classical Hensel's lemma asserts that if polynomials F(x), G_{0}(x), H_{0}(x) in R_{v}[x] are such that (i) v^{0}(F(x) – G_{0}(x)H0(x)) > 0, (ii) the leading coefficient of G_{0}(x) has v-valuation zero, (iii) there are polynomials A(x), B(x) belonging to the valuation ring of v^{x} satisfying v_{x}(A(x)G_{0}(x) + B(x)H_{0}(x) – 1) > 0, then there exist G(x), H(x) in K[x] such that (a) F(x) = G(x)H(x), (b) deg G(x) = deg G_{0}(x), (c) v^{x}(G(x)–G_{0}(x)) > 0, v^{x}(H(x) – H_{0}(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 wa transcendental extension of k_{v}.

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ID Code: | 69891 |

Deposited On: | 19 Nov 2011 10:25 |

Last Modified: | 19 Nov 2011 10:25 |

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