Extension d'une valuation
Extension d'une valuation
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延伸估价
DOI：
10.1090/s0002-9947-07-04184-0
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发表时间：
2007
影响因子：
1.3
通讯作者：
M. Vaquié
中科院分区：
文献类型：
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作者：
M. Vaquié
我们要确定域K的赋值v到K的循环扩张L的所有扩张，即L = K（x）是x的有理函数域或L = K（θ）是由不可约多项式G（x）的根0生成的有限可分扩张。在1936年的两篇文章中，桑德斯·麦克莱恩（Saunders MacLane）对K[x]的给定赋值μ引入了关键多项式和增广赋值的概念，并展示了如何通过可数增广赋值序列（μ m）i ∈ I <$N恢复K的离散秩1赋值v对L的任何扩展。赋值μ i是由赋值μ i-1、一个关键多项式o i和值γ i = μ（oi）的归纳定义的。本文研究了增广赋值的一些性质，并将MacLane的结果推广到K的任意赋值v的情形。为此，我们需要引入简单的容许增广赋值族A（μα）α ∈A（其中A不一定是可数集），并定义这类族的极限关键多项式和极限增广赋值.那K上赋值ν到L的任何扩张μ也是增广赋值族的极限。我们还得到了一个“因子分解”定理，它给出了K [x]中任意多项式f的值（μ α（f））的描述。
We want to determine all the extensions of a valuation v of a field K to a cyclic extension L of K, i.e. L = K(x) is the field of rational functions of x or L = K(θ) is the finite separable extension generated by a root 0 of an irreducible polynomial G(x). In two articles from 1936, Saunders MacLane has introduced the notions of key polynomial and of augmented valuation for a given valuation μ of K[x], and has shown how we can recover any extension to L of a discrete rank one valuation v of K by a countable sequence of augmented valuations (μ ι ) ι ∈ with I ⊂ N. The valuation μ i is defined by induction from the valuation μ i-1 , from a key polynomial o i and from the value γ, = μ(oi). In this article we study some properties of the augmented valuations and we generalize the results of MacLane to the case of any valuation v of K. For this we need to introduce simple admissible families of augmented valuations A (μα) α ∈A where A is not necessarily a countable set, and to define a limit key polynomial and limit augmented valuation for such families. Then. any extension μ to L of a valuation ν on K is again a limit of a family of augmented valuations. We also get a "factorization" theorem which gives a description of the values (μ α (f)) for any polynomial f in K [x] .