Introduction to ordered fields

Let \(K\) be a field. An ordering of \(K\) is a subset \(P\) of \(K\) having the following properties:

ORD 1

Given \(x \in K\), we have either \(x \in P\), or \(x=0\), or \(-x \in P\), and these three possibilities are mutually exclusive. In other words, \(K\) is the disjoint union of \(P\), \(\{0\}\), and \(-P\).

ORD 2

If \(x, y \in P\), then \(x+y\) and \(xy \in P\).

We shall also say that \(K\) is ordered by \(P\), and we call \(P\) the set of positive elements. Continue reading An infinite field that cannot be ordered →