An algebraically closed field
An algebraically closed field
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代数闭域
DOI：
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发表时间：
1968
影响因子：
0.5
通讯作者：
F. Rayner
中科院分区：
文献类型：
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作者：
F. Rayner
设k是任意代数闭域，k((T))表示k上一个不定t中的形式幂级数的域，使得K是系数在k中的Puiseux展开式的域(K的每个元素都是tl/r中的形式幂级数，对于某个正整数r).众所周知，K是代数闭的当且仅当k具有特征零[1，p.61]。对于赋值域的分支扩张的例子[9，§6]，当k具有非零特征p时，有一个类似于K的域是代数闭的是有用的。在这篇文章中，我证明了形式为Σaitei(其中(Ei)是良序的，Ei=mi|nprt，n∈Ζ，mi∈Ζ，ai∈k，ri∈Ν)的所有形式幂级数的集合L形成了一个代数闭域。
Let k be any algebraically closed field, and denote by k((t)) the field of formal power series in one indeterminate t over k. Let so that K is the field of Puiseux expansions with coefficients in k (each element of K is a formal power series in tl/r for some positive integer r). It is well-known that K is algebraically closed if and only if k is of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous to K which is algebraically closed when k has non-zero characteristic p. In this paper, I prove that the set L of all formal power series of the form Σaitei (where (ei) is well-ordered, ei = mi|nprt, n ∈ Ζ, mi ∈ Ζ, ai ∈ k, ri ∈ Ν) forms an algebraically closed field.