Bhatia, Saurabh ; Khanduja, Sudesh K.
(2002)
*On extensions generated by roots of lifting polynomials*
Mathematika, 49
(1-2).
pp. 107-118.
ISSN 0025-5793

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

Related URL: http://dx.doi.org/10.1112/S0025579300016107

## Abstract

Let v be a Henselian valuation of any rank of a field K and ν̅ its unique prolongation to a fixed algebraic closure K̅ of K having value group G̅. For any subfield L of K̅ ,let R(L) denote the residue field of the valuation obtained by restricting ν̅ to L. Using the canonical homomorphism from the valuation ring of v onto its residue field R(K), one can lift any monic irreducible polynomial with coefficients in R(K) to yield a monic irreducible polynomial with coefficients in K. In an attempt to generalize this concept, Popescu and Zaharescu introduced the notion of lifting with respect to a (K, v)-minimal pair (α, δ) belonging to K̅ × G̅. As in the case of usual lifting, a given monic irreducible polynomial Q(y) belonging to R(K(α))[y] gives rise to several monic irreducible polynomials over K which are obtained by lifting with respect to a fixed (K, v)-minimal pair (α, δ). If F, F _{1} are two such lifted polynomials with coefficients in K having roots θ, θ_{1}, respectively, then it is proved in the present paper that ν̅(K(θ)= ν̅ (K(θ_{1})),R(K(θ))= R(K(θ_{1}))in case (K, v) is a tame field, it is shown that K(θ) and K(θ1) are indeed K-isomorphic.

Item Type: | Article |
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Keywords: | 12E05: Field Theory and Polynomials: General Field Theory; Polynomials (irreducibility; etc) |

ID Code: | 69890 |

Deposited On: | 19 Nov 2011 11:05 |

Last Modified: | 19 Nov 2011 11:05 |

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