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- Title
Prolongations of valuations to finite extensions.
- Authors
Khanduja, Sudesh K.; Kumar, Munish
- Abstract
Let K = ℚ(θ) be an algebraic number field with θ in the ring AK 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 f̄ (x) = ḡ1(x)e1 …. ḡr (x)er be the factorization of the polynomial f̄ (x) obtained by reducing coefficients of f (x) modulo p into a product of powers of distinct irreducible polynomials over ℤ/pℤ with gi (x) monic. Dedekind proved that if p does not divide [AK : ℤ [θ]], then pAK = ℘1e1 … ℘rer, where ℘1, …, ℘r are distinct prime ideals of AK, ℘i = pAK +gi (θ)AK having residual degree equal to the degree of ḡi (x). He also proved that p does not divide [AK : ℤ[θ]] if and only if for each i, either ei = 1 or ḡi (x) does not divide M̄(x) where M(x) = 1/p (f (x) - g1(x)e1 …. gr (x)er). Our aim is to give a weaker condition than the one given by Dedekind which ensures that if the polynomial f̄ (x) factors as above over ℤ/pℤ, then there are exactly r prime ideals of AK lying over p, with respective residual degrees deg ḡ1(x), …, deg ḡr (x) and ramification indices e1, …, er. 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.
- Subjects
ALGEBRAIC fields; FINITE fields; RINGS of integers; RATIONAL numbers; FACTORIZATION; POLYNOMIALS
- Publication
Manuscripta Mathematica, 2010, Vol 131, Issue 3/4, p323
- ISSN
0025-2611
- Publication type
Article
- DOI
10.1007/s00229-009-0320-1