# Prolongations of valuations to finite extensions

Khanduja, Sudesh K. ; Munish, Kumar (2010) Prolongations of valuations to finite extensions Manuscripta Mathematica, 131 (3-4). pp. 323-334. ISSN 0025-2611

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Related URL: http://dx.doi.org/10.1007/s00229-009-0320-1

## Abstract

Let K=Q(θ) be an algebraic number field with θ in the ring A K of algebraic integers of K and f(x) be the minimal polynomial of θ over the field Q of rational numbers. For a rational prime p, let f̅(x) = g̅1(xθ1)...gr(x)θr be the factorization of the polynomial f̅(x) obtained by reducing coefficients of f(x) modulo p into a product of powers of distinct irreducible polynomials over Z/PZ with g i (x) monic. Dedekind proved that if p does not divide [ AK:Z [θ]], then pAK=℘1θ1...,℘rθr where ℘1,...,℘r are distinct prime ideals of A K , ℘i=pAK+gi(θ)AK having residual degree equal to the degree of g̅1(x) . He also proved that p does not divide [ AK:Z [θ]] if and only if for each i, either e i = 1 or g̅1(x) does not divide M̅ where M(x)=1÷p (f(x)-g1(x)θ1....gr(x)θr). Our aim is to give a weaker condition than the one given by Dedekind which ensures that if the polynomial f̅(x) factors as above over Z/pZ, then there are exactly r prime ideals of A K lying over p, with respective residual degrees deg g̅1(x),..., deg g̅r(x)and ramification indices e 1, ..., e r . In this paper, the above problem has been dealt with in a more general situation when the base field is a valued field (K, v) of arbitrary rank and K(θ) is any finite extension of K.

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