Prolongations of valuations to finite extensions

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(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 g̅₁(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 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 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.