In following post, I describe the "classical" example of $\mathbb{Q}(\sqrt{2})$ that can be ordered in two distinct ways.

More generally, if $(k,P)$ is an ordered field, $R$ a real closure of $(k,P)$ and $K/k$ an algebraic extension, how can we describe the orders of $K$ that extend $P$?

Can one produce a case where the number of orders of $K$ that extends $P$ is infinite?

In following post, I describe the "classical" example of $\mathbb{Q}(\sqrt{2})$ that can be ordered in two distinct ways.

More generally, if $(k,P)$ is an ordered field, $R$ a real closure of $(k,P)$ and $K/k$ an algebraic extension, how can we describe the orders of $K$ that extend $P$?

Can one produce a case where the number of orders of $K$ that extend $P$ is infinite?