Bishnoi, Anuj ; Kumar, Sanjeev ; Khanduja, Sudesh K.
(2013)
*On liftings of powers of irreducible polynomials*
Journal of Algebra and Its Applications, 12
(05).
Article ID 1250222.
ISSN 0219-4988

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Official URL: http://www.worldscientific.com/doi/abs/10.1142/S02...

Related URL: http://dx.doi.org/10.1142/S0219498812502222

## Abstract

Let v be a henselian valuation of arbitrary rank of a field K with valuation ring Rv having maximal ideal Mv. Using the canonical homomorphism from Rv onto Rv/Mv, one can lift any monic irreducible polynomial with coefficients in Rv/Mv to yield monic irreducible polynomials over Rv. Popescu and Zaharescu extended this approach and introduced the notion of lifting with respect to a residually transcendental prolongation w of v to a simple transcendental extension K(x) of K. As it is well known, the residue field of such a prolongation w is L̅(Y), where L̅ is the residue field of the unique prolongation of v to a finite simple extension L of K and Y is transcendental over L̅ (see [V. Alexandru, N. Popescu and A. Zaharescu, A theorem of characterization of residual transcendental extension of a valuation, J. Math. Kyoto Univ.28 (1988) 579–592]). It is known that a lifting of an irreducible polynomial belonging to L̅[Y] with respect to w, is irreducible over K. In this paper, we give some sufficient conditions to ensure that a given polynomial in K[x] satisfying these conditions which is a lifting of a power of some irreducible polynomial belonging to L̅[Y] with respect to w, is irreducible over K. Our results extend Eisenstein–Dumas and generalized Schönemann irreducibility criteria.

Item Type: | Article |
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Source: | Copyright of this article belongs to World Scientific Publishing Company. |

Keywords: | Valued Fields; Non-Archimedean Valued Fields; Irreducible Polynomials |

ID Code: | 112138 |

Deposited On: | 23 Jan 2018 12:19 |

Last Modified: | 23 Jan 2018 12:19 |

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