Questions tagged [ordered-fields]

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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 ...
509 views

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 ...
607 views

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 $F$-...
313 views

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 ...
117 views

'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 ...
243 views

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 ...
96 views

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 ...
235 views

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 ...