A field K with an archimedean valuation is ismorphic to a subfield of C and is equivalent to the absolut ordinary valuation in C

Question

I've found this statement in the book I'm using for my final project. I don't seem to be able to find a demonstration for it. The closest I've found is in the book Local Fields by J.W.S. Cassels, but it's demonstrated for complete fields with an archimedean valuation, and not all fields.

The original Theorem is in french:

Thanks in advance to anyone that can help me!

Answer 1

The general case follows immediately from the complete case, by taking the completion. If $K$ is a an archimedean valued field, then its completion is a complete archimedean valued field, which is then isomorphic (as a valued field) to a subfield of $\mathbb{C}$. Since $K$ embeds in its completion, the same is true of $K$.