## 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