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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Official URL: http://www.springerlink.com/content/m1568w46pq3334...

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) = g̅_{1}(x^{θ1}^{)...}g_{r}(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 [ A_{K}:Z [θ]], then pA_{K}=℘_{1}^{θ1}^{...},℘_{r}^{θr} where ℘_{1},^{...},℘_{r} are distinct prime ideals of A K , ℘_{i}=pA_{K}+g_{i}(θ)A_{K} having residual degree equal to the degree of g̅_{1}(x) . He also proved that p does not divide [ A_{K}:Z [θ]] if and only if for each i, either e i = 1 or g̅_{1}(x) does not divide M̅ where M(x)=1÷p
(f(x)-g_{1}(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 g̅_{1}(x),^{...}, deg g̅_{r}(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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Deposited On: | 19 Nov 2011 10:27 |

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