Residual ideals of MacLane valuations
Residual ideals of MacLane valuations
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麦克莱恩估值的剩余理想
DOI：
10.1016/j.jalgebra.2014.12.022
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
E. Nart
中科院分区：
文献类型：
--
作者：
Julio Fern'andez;J. Guàrdia;J. Montes;E. Nart
设（K，v）是一个离散值域，x是一个不定值.在1936年，麦克莱恩确定了所有的估值K（x）扩展v.他的工作已审查和推广瓦奎，通过使用分次代数的估值。我们延长瓦奎的方法通过研究剩余理想的分级代数作为一个抽象的对应物的某些剩余多项式具有关键作用的理论在计算应用。这使我们能够确定K（x）上离散赋值的分次代数的结构。设P是v的赋值环的完备化O v中系数的一元不可约多项式的集合。本文的构造方法在素多项式的Okutsu等价类的集合和某个MacLane空间M之间产生一个规范双射P/M → M，其点可以用与K（x）上的赋值相关的离散参数来描述。
Let (K, v) be a discrete valued field and let x be an indeterminate. In 1936, MacLane determined all valuations on K (x) extending v. His work has been reviewed and generalized by Vaquié, by using the graded algebra of a valuation. We extend Vaquié's approach by studying residual ideals of the graded algebra as an abstract counterpart of certain residual polynomials having a key role in the computational applications of the theory. This enables us to determine the structure of the graded algebra of the discrete valuations on K (x). Also, let P be the set of monic irreducible polynomials with coefficients in the completion O v of the valuation ring of v. The constructive methods of the paper yield a canonical bijection P/≈→ M, between the set of Okutsu equivalence classes of prime polynomials and a certain MacLane space M, whose points may be described in terms of discrete parameters associated with valuations on K (x).