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 →