On the extensions of discrete valuations in number fields
-
Abdulaziz Deajim
and
Lhoussain El Fadil
Abstract
Let K be a number field defined by a monic irreducible polynomial F(X) ∈ ℤ [X], p a fixed rational prime, and νp the discrete valuation associated to p. Assume that F(X) factors modulo p into the product of powers of r distinct monic irreducible polynomials. We present in this paper a condition, weaker than the known ones, which guarantees the existence of exactly r valuations of K extending νp. We further specify the ramification indices and residue degrees of these extended valuations in such a way that generalizes the known estimates. Some useful remarks and computational examples are also given to highlight some improvements due to our result.
(Communicated by Filippo Nuccio)
Acknowledgement
The authors are deeply grateful to the anonymous referee whose valuable comments and suggestions have tremendously improved the quality of the paper. A. Deajim would like to thank the University Council and the Scientific Council of King Khalid University for approving a sabbatical leave request for the academic year 2018–2019, during which this article was finalized and submitted. He also expresses his gratitude to King Khalid University for providing administrative and technical support. L. El Fadil is very grateful to Professor Enric Nart for introducing him to the techniques of Newton polygons when he was a post-doc in 2007–2008 at CRM, Barcelona, Spain.
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© 2019 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
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- Natural extension of EF-spaces and EZ-spaces to the pointfree context
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- On the extensions of discrete valuations in number fields
- On certain classes of generalized derivations
- Nilpotent elements in medial semigroups
- Power of meromorphic function sharing polynomials with derivative of it’s combination with it’s shift
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Articles in the same Issue
- Regular papers
- Professor Anatolij Dvurečenskij
- A generating theorem of punctured surface triangulations with inner degree at least 4
- Natural extension of EF-spaces and EZ-spaces to the pointfree context
- On weak isometries in directed groups
- On self-injectivity of the f-ring Frm(𝓟(ℝ), L)
- On the extensions of discrete valuations in number fields
- On certain classes of generalized derivations
- Nilpotent elements in medial semigroups
- Power of meromorphic function sharing polynomials with derivative of it’s combination with it’s shift
- Sharp coefficient bounds for starlike functions associated with the Bell numbers
- Sufficient conditions for Carathéodory functions and applications to univalent functions
- Blending type approximation by complex Szász-Durrmeyer-Chlodowsky operators in compact disks
- Linear directional differential equations in the unit ball: solutions of bounded L-index
- On oscilatory fourth order nonlinear neutral differential equations – IV
- Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms
- Lerch’s theorem on nabla time scales
-
- Convolution theorems related with the solvability of Wiener-Hopf plus Hankel integral equations and Shannon’s sampling formula
- Stabilization of third order differential equation by delay distributed feedback control with unbounded memory
- On the amenability of a class of Banach algebras with application to measure algebra
- A bivariate Kumaraswamy-exponential distribution with application
- Berry-Esseen bounds for wavelet estimator in time-varying coefficient models with censored dependent data
- Corrigendum to ”on the congruent number problem over integers of cyclic number fields”, Math. Slovaca 66(3) (2016), 561–564
