Deveno, why would you go to such horrific lengths to come up with some monstrosity (your post #29) you claim is a field which has a “member” such that “member”^2 = 2 and is not Archimedean (a pure field). It’s a total waste since fields are isomorphic.
Here’s how its done (reference my post #28):
The field of rationals is complete because infinite denominators are included in the field. Thereforee x^2=2 has a rational solution. (No Archimedean "Postulate")