Problem about ordered fields

Question

Let $K_0$, $K_1$, $K_2$ be ordered fields and $i_k$:$K_0$ $\rightarrow K_k$ are ordered inclusions for $k=1,2$. Show that there exists ordered field $K$ and ordered insclusions $j_k$ : $K_k \rightarrow K$ for which $j_1 \circ i_1 = j_2 \circ i_2$

Since i have not much knowledge about ordered fields would appreciate any help or ideas about this problem.

Answer 1

I am sorry I forgot to actually propose an answer.

Let $F$ be a $\kappa$-saturated ordered field where $\kappa >|K_1|,|K_2|$ (for instance a ultrapower of $\mathbb{R}$ modulo a free ultrafilter on $\max(|K_1|,|K_2|)$).

Thus there is an embedding $j_1: K_1 \rightarrow F$. Let $j_2'$ be the embedding $i_2(K) \rightarrow F$ satisfying $j_2' \circ i_2= j_1 \circ i_1$. Since $F$ is $|K_2|^+$-saturated, the morphism $j_2'$ extend to $K_2$, giving a solution $j_2$ to the almagamation problem.