# Questions tagged [ordered-fields]

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### Constructing ordered fields with lattice structure from ordered fields without lattice structure, and vice versa, in constructive mathematics

This post originated from my reference request for the definition of an ordered field in constructive mathematics: Proper definition of ordered field in constructive mathematics We are working in ...
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### Witt ring of a field with Pythagoras number $2$

I am currently looking at a few simple properties of the Witt ring of a field $K$ (by which I mean the ring of Witt classes of quadratic forms, not the ring of Witt vectors), which are clearly true ...
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### Reduced power of an ordered field

Suppose $K$ is an ordered field, $X$ is any set, and $F$ is a filter over $X$. Let $G$ be the reduced power of $K$ by $F$. That is, we take all functions from $X$ to $K$, then take equivalence ...
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### Is this theory the complete theory of the real ordered field?

We know that the real ordered field can be characterized up to isomorphism as a complete ordered field. However this is a second order characterization. That raises the following question. Consider ...
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### Is there a complete characterization of ordered fields without definable proper subfields?

$\mathbb{Q}$ has no proper subfields. As a result, all ordered fields elementarily equivalent to $\mathbb{Q}$ have no proper subfields which are first-order definable without parameters. And by the ...
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### Request for bibliographic information

Greetings to everyone on this forum (I am a new-comer). I would like to ask the experienced members for suggestions on (as) comprehensive and systematic (as possible) bibliographic sources regarding: ...
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### Decidability of a first-order theory of hyperreals

The theory of real closed fields is decidable. The hyperreals satisfy that theory, so we can interpret statements in the theory of real closed fields as being about hyperreals. If we add a unary ...
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### Tensor product of preordered rings

All rings in this post are commutative, unital, and contain $\frac{1}{2}$. To study "real" properties of a ring $R$, one is often interested in the orderings which exist on fraction fields of ...
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### Are radicals dense in the real closure of an ordered field?

Let $F$ be an ordered field and let $R$ denote its real closure. It is well-known that $F$ is cofinal in $R$, but not necessarily dense. For example, consider $F=\mathbb{R}(\omega)$ with the order ...
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### Cauchy reals and Dedekind reals satisfy "the same mathematical theorems"

The succinct question The conjecture of Birch and Swinnerton-Dyer (to take a random example) mentions L-functions and hence the complex numbers and hence the real numbers (because the complexes are ...
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### Are archimedean subextensions of ordered fields dense?

Let $E$ be an ordered field and let $F$ be a real closed subfield. We say that $E$ is $F$-archimedean if for each $e\in E$ there is $x\in F$ such that $-x\le e\le x$. Is it true that if $E$ is $F$-...
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### How do fractional tensor products work?

[I asked and bountied this question on Math SE, where it got several upvotes and a comment suggesting it was research-level, but no answers. So I'm reposting here with slight edits, but please feel ...
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### 'Smallest' subfield of the Surreals which is isomorphic to the Surreals as an ordered group

What is the smallest subfield $F\subset N_0$ such that $$(F,+,\times,\leq)\ncong(N_0,+,\times,\leq)$$ but $$(F,+,\leq)\cong(N_0,+,\leq)?$$ Since these are all going to be proper classes cardinality is ...
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### Formally real fields with unique non-Archimedean ordering

My question is rather simple. Do there exist a formally real field that admits a unique ordering (so sums of squares are the positive elements) and such that this ordering is not archimedean? Oh, I ...
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### Atomic integer parts

Let $R$ be an ordered ring (in particular, the order is linear and $R$ is a domain). Let $| \ \ |: x \mapsto \max(x,-x)$ denote its absolute value. For $(x,y) \in R \times R^{\neq 0}$ say that an ...
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### Do all fields with internal absolute values arise as ordered fields or like $\mathbb{C}$ from them?

$\def\abs#1{\lvert#1\rvert} \def\Im{\operatorname{Im}} \def\Re{\operatorname{Re}}$ (Crossposted from math.stackexchange.com after 5 days with no correct answer.) Let ​ $\langle F,+,\cdot\rangle$ ​ be ... 1 vote
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### analysis over non-Archimedean ordered fields

Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...
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