How to prove that this enough to determine the structure of $\mathbb R$?

Question

Let $F$ be a field and let $P\subset F$ such that the sets $P$, $\left\{ 0\right\} $ and $-P:=\left\{ -p:p\in P\right\} $ form a partition of $F$ and $x,y\in P\Rightarrow x+y\in P\wedge xy\in P$.

I know that the relation $<$ on $F$ defined by $x<y\iff y-x\in P$ makes $F$ an ordered field. If next to that $\left(F,<\right)$ has the least upper bound property (i.e. every non-empty set $A\subset F$ bounded above has a least upper bound) then it can be proved that $\left(F,<\right)$ and $\left(\mathbb{R},<\right)$ are ismorphic when it comes to order respecting ringhomomorphisms.

Actually I am looking for a concise proof of that fact.

Thanks in advance.

Answer 1

Have in mind that you must show the following:

1. Prove that $F$ has a subfield $K$ isomorphic to $\Bbb Q$. You will need that $1+1>0$ to guarantee that the charasteristic of $F$ is $0$.
2. $\Bbb R$ is precisely the set of the suprema of all bounded sets of $\Bbb Q$. Create a similar structure in $F$ and build the isomorphism. Perhaps (I am not sure now) you need the fact that $\Bbb Q$ is dense in $\Bbb R$.

This is not exactly concise, though...