Every totally ordered field satisfying archimedean property can be embedded in the real numbers.

Question

I need a reference for the following result:

Theorem: Every totally ordered field satisfying the Archimedean property can be embedded in the real numbers.

There are books that mention this result without any proof or reference. Someone knows any good reference? The more the better.

Answer 1

I found a very nice paper that deals with different (equivalent) characterizations of the real numbers. A good reference for the result in question is Theorem 3.5 of Hall, James Forsythe. "Completeness of ordered fields." arXiv preprint arXiv:1101.5652 (2011).

Answer 2

The basic idea is fairly simple: let $F$ be such a field. Then you observe

• There are no infinitesimals in $F$
• The rationals are dense in $F$

and apply in some fashion the fact that the field of reals is the completion of the field of rationals.

Answer 3

Archimedean ordered fields are real looks good to me. It's a proof of the proposition using Dedekind cuts.