Valuations in algebraic field extensions

JOURNAL OF ALGEBRA

卷:312

Article

Govantes, F. J. Herrera ; Acosta, M. A. Alla ; Spivakovsky, M.

关键词: valuation;    algebraic extension;    key polynomial;    Newton polygon;   

DOI  :  10.1016/j.jalgebra.2007.02.022

来源: Elsevier

【 摘 要 】

Let K -> L be an algebraic field extension and v a valuation of K . The purpose of this paper is to describe the totality of extensions {v'} of v to L using a refined version of MacLane's key polynomials. In the basic case when L is a finite separable extension and rk v = 1, we give an explicit description of the limit key polynomials (which can be viewed as a generalization of the Artin-Schreier polynomials). We also give a realistic upper bound on the order type of the set of key polynomials. Namely, we show that if char K = 0 then the set of key polynomials has order type at most N, while in the case char K = p > 0 this order type is bounded above by ([log (p)n] + 1)omega, where n = [L : K]. Our results provide a new point of view of the well-known formula Sigma(S)(j=1), e(j)f(j)d(j) = n and the notion of defect. (c) 2007 Elsevier Inc. All rights reserved.

【 授权许可】

Free