I am looking for a reference on work regarding the order types of ordered fields. I am particularly interested in known caracterisations of such order types in the $2^{nd}$-order language of ordered sets, if there are such caracterisations.
Those orders $(O,<)$ have to be non-empty, total, dense, without extremities and they have to satisfy $(i)$: $\forall x < y \in O$, $O$ and $O^*$ are isomorphic to $]x;y[$.
Those are strong conditions, but they are unlikely sufficient.
So far, not knowing what is known about this problem, I have focused my efforts on figuring whether under the following additionnal condition:
$(ii)$: $(O,<)$ is not $(cf(O), cf(O))$ complete, in the sense that there is an unfilled Dedekind cut $(A \ | \ B)$ in $O$ where $A \cong B^* \cong cf(O)$.
there is an order isomorphism from $O$ to $\mathbb{Q}(O)$ where $\mathbb{Q}(O)$ is the ordered field of fractions with rationnal coefficients, indeterminates $X_o, o\in O$ in $O$, and where $\mathbb{Q}[X_o] < X_{o'}$ whenever $o < o'$.
Assumption $(ii)$ allows one to prove that $O$ is isomorphic to any $O - \{o\}, o \in O$, which may or may not be a good starting point to find an isomorphism. We can see that for $O = \mathbb{R}$, $(ii)$ fails and $\mathbb{R}$ is not isomorphic to $\mathbb{Q}(\mathbb{R})$ since the former is Dedekind complete while the latter is not.
Does somebody know of a reference where this topic is discussed? Does anyone have answers to my problems regarding the sufficiency of $(i)$ and $(ii)$ for each purpose?