Galois extensions and O*-fields

A field F is O* if each partial order that makes F a partially ordered field can be extended to a total order that makes F a totally ordered field. We use the theory of infinite primes developed by Dubois and Harrison to prove the following. For a subfield F of C that is finite-dimensional over Q, we prove that when F is Galois over Q, F is an O*-field if and only if is a subfield of R. We find other conditions that make F an O*-field and provide several examples. As well for an arbitrary field of characteristic 0, we characterize the maximal partial orders that are Archimedean.