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Generalized Hensel's lemma

Published online by Cambridge University Press:  20 January 2009

Sudesh K. Khanduja  and
Jayanti Saha
Sudesh K. Khanduja
Affiliation:
Centre for Advanced Study in Mathematics, Panjab UniversitySector – 14 Chandigarh – 160014, India
Jayanti Saha
Affiliation:
Centre for Advanced Study in Mathematics, Panjab UniversitySector – 14 Chandigarh – 160014, India
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Let (K, v) be a complete, rank-1 valued field with valuation ring Rv, and residue field kv. Let vx be the Gaussian extension of the valuation v to a simple transcendental extension K(x) defined by The classical Hensel's lemma asserts that if polynomials F(x), G0(x), H0(x) in Rv[x] are such that (i) vx(F(x) – G0(x)H0(x)) > 0, (ii) the leading coefficient of G0(x) has v-valuation zero, (iii) there are polynomials A(x), B(x) belonging to the valuation ring of vx satisfying vx(A(x)G0(x) + B(x)H0(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 G0(x), (c) vx(G(x)–G0(x)) > 0, vx(H(x) – H0(x)) > 0. In this paper, our goal is to prove an analogous result when vx is replaced by any prolongation w of v to K(x), with the residue field of wa transcendental extension of kv.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 42 , Issue 3 , October 1999 , pp. 469 - 480
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

REFERENCES

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