Definition 1: A field $F$ is an order field if
a) $x,y,z\in F,y \lt z \to x + y \lt x + z$
b) $x,y\in F, x \gt 0, y \gt 0 \to xy \gt 0$
Definition 2: A field $F$ is called an ordered field if there exists $P \in F$ s.t.
a) $x,y \in P \to x+y \in P, xy \in P$
b) $x \in F\to x \in P$ or $x=0$ or $-x\in P$
I think a logical path would be to use the properties of a field to derive the set P from the conditions in definition 1. However, I'm not really sure if that is entirely correct or even how to go about it. Any help would be appreciated.
Hint: For the implication $(1) \implies (2)$, show that the set $P=\{x \in F \mid x>0\}$ gives you what you want in $(2)$.
For the implication $(2) \implies (1)$, define $<$ by saying that $x<y$ if $y-x \in P$, then show that $<$ is a strict total order which satisfies the conditions of $(1)$.
To understand the intuition, the idea is that $P$ is the set of positive elements.
(Also, strictly speaking, in definition $(2)$ the set $P$ should be supplied from the get-go (in the same way $<$ is provided in definition $(1)$). Otherwise I'd say that $F$ isn't ordered, but is orderable since a priori there's no reason to suspect $P$ is unique.)