# Questions tagged [ordered-fields]

Ordered fields are fields which have an additional structure, a linear order compatible with the field structure. This tag is for questions regarding ordered fields and their properties, as well proofs related to un-orderability of certain fields.

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### Is every ordered field a topological field?

Let $F$ be an ordered field and give $F$ an order topology. Then is $F$ a topological field (that is, are the operations \begin{equation} \begin{split} +&:F\times F\to F\\ -&:F\to F\\ \times&...
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### A natural choice of 'maximal ordered subfield' of a field?

For any field $K$ of characteristic zero, its prime subfield $Q(K)$ has a natural ordering inherited from $\mathbb{Q}$. I am interested in finding out to what extent (if at all) this natural ordering ...
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### Related to partially ordered field (poset) and total ordered field

(i) Let K be a field with the property that there is a positive natural number m such that $m * 1 = 0$. Show that there is no total order that gives K the structure of an ordered field. (ii) Show that ...
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### $F(\sqrt x \mid x > 0)$ is formally real

In Jacobson's Basic Algebra IV, it is claimed in the proof of Theorem 8 page 285 that if $F$ is an ordered field then the extension $L = F(\sqrt x\mid x > 0)$ obtained by adjoining the square roots ...
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### is this proof of 1>0 in an ordered field correct?

As I do not know what to write here, I write only that I tried to prove that $1>0$ using the ordered field axioms (Ordered Field Axioms). I simplified what the site said for shortness only. Try ...
1answer
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### Is there a way to represent the relation $<$ on the real numbers without using multiplication?

I know for $x,y\in\mathbb R$ we have $x<y$ iff there exists a $z\in\mathbb R$, so that $y=x+z^2$. Is there a similar way to „prove“ $x<y$ using only $+$ and $=$? And if not, is there a more or ...
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### What is a connex relation?.

I've been recently trying to teach myself some topology and in the book I'm reading there is the definition of an order relation which confuses me a lot . I've looked online for more information and ...
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### Solving conjugation equations in bi-ordered groups

A bi-ordered group is a group $G$ equipped with a partial order $<$ with: $f<g\Longleftrightarrow h f i < h g i$ for all $f,g,h,i \in G$. In particular this is the case for groups of strictly ...
1answer
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### How to prove: Any positive real number power of positive real number is positive.

I have a question regarding how to prove: Any positive real number power of a positive real number is positive. I have shown that any positive rational power of positive real numbers is positive. I ...
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### Is there an ordered field with distinct subfields isomorphic to the reals?

Is there an ordered field with distinct subfields isomorphic to the field $\mathbb R$ of real numbers?
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### Fields non-trivially seen to be isomorphic to the real numbers by the uniqueness of a complete ordered field

It is a well-known fact from introductory real analysis that $\mathbb{R}$ is the unique Dedekind-complete ordered field, up to isomorphism. Hence, we are in a sense justified in an abstract definition ...