For a short reminder about ordered fields you can have a look to following post. We prove there that $\mathbb{Q}$ can be ordered in only one way.

That is also the case of $\mathbb{R}$ as $\mathbb{R}$ is a real-closed field. And one can prove that the only possible positive cone of a real-closed field is the subset of squares.

However $\mathbb{Q}(\sqrt{2})$ is a subfield of $\mathbb{R}$ that can be ordered in two distinct ways. Continue reading A field that can be ordered in two distinct ways →

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 →