关于有序域的几则短记

Dubois在文[1]中举反例示明,在Artin所解决的Hilbert第17问题中所出现的系数域是个亚基米德有序域这一条件是不能减弱的(见[1]或者[7])在他所举出的反例中,隐约地使用了一些有关序与赋值相互关系的概念与事实,但都没有明显地提出或证明。这篇短文首先是阐明隐含在[1]中的一些概念和事实;其次指出他的例子还可以用来说明

Some Remarks on Ordered Fields

In this note, Dubois' counter-example to Lang's formulation of Hilbert's 17-th problem is re-examined, besides, a field admitting only one ordering is constructed which has a non-henselian valuation ring compatible with its ordering. Let F=Q(?), let t be an indeterminate, let F(t) be ordered so that t is posi tive and infinitesimal, and let K be the quadratic closure of F(t) with respect to the given ordering. The canonical valuation ring A of K is of course compatible with the unique ordering of K. However, it is shown that A is not henselian.