1
vote
1answer
18 views

When can complete dense linear orders be made into topological fields?

By a "complete dense linear order," I mean a dense linear order in which every nonempty subset with an upper bound has a least upper bound. The canonical example, $\mathbb{R}$, is a topological field ...
2
votes
2answers
95 views

Embedding ordered number fields into $\Bbb R$

Consider a number field $K$ equipped with a total order $\le$. Is there always a field homomorphism $\phi:K \rightarrow \Bbb R$ which respects the total order, i. e. with $\phi(x)>\phi(y)$ for ...
0
votes
1answer
146 views

What is a valuation associated to an ordering on a field?

If $(K,\leq)$ is a totally ordered field with $P\!=\!\{\alpha\!\in\!K;\, 0\!\leq\!\alpha\}$, how is the valuation associated to $P$ defined? I was searching through Prestel & Delzell's Positive ...
5
votes
3answers
804 views

Isomorphisms: preserve structure, operation, or order?

Everyone always says that isomorphisms preserve structure... but given the (multiple) definitions of isomorphism, I fail to see how the definitions equate with the intuitive meaning, which is that two ...
2
votes
1answer
203 views

algebraic closure and real closure are closure operators?

Are the algebraic closure (of a field) and the real closure (of a totally ordered field) closure operators (when restricted to appropriate sets of fields, so that they are maps on a set instead of a ...
10
votes
2answers
197 views

Is it possible to construct an ordered field which is also algebraically closed?

It is well known that while the real numbers are totally ordered, they are not algebraically closed, and while the complex numbers are algebraically closed, they are not totally ordered. Is it ...
5
votes
2answers
247 views

Is $\mathbb R$ terminal among Archimedean fields?

I was wondering why metrics and norms are always defined to be real, rather than generalized to some other fields (or whatever). The best guess I have so far is: Because every Archimedean ordered ...
5
votes
1answer
731 views

Least upper bound property iff convergence of Cauchy sequences

From http://en.wikipedia.org/wiki/Non-Archimedean_ordered_field: The field of rational functions over $\mathbb{R}$ can be used to construct an ordered field which is complete (in the sense of ...
1
vote
1answer
79 views

Is there a general method to order an arbitrary field extension?

Here is something I've been wondering about recently. Suppose you have an arbitrary ordered field $F$, and let $F(\sqrt{a})$ be a field extension with $a>0$ in $F$. Is there then some way to order ...