>>40
I haven't studied abstract algebra yet, but the ordered field definition I'm aware of does not rely on the Archimedean property as an axiom, and as was shown in >>30 this is provable as a theorem and does not need to be an axiom (this is not from the same book but I'm fairly certain these are the same properties):
http://math.colorado.edu/~packer/Orderedfieldaxioms300120.pdf