# 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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### Does every partially ordered commutative ring admit an order preserving homomorphism to a real closed field?

I hope the answer is yes. It would suffice to construct a homomorphism to a formally real field. I think it would then suffice to extend the partial order to a total ring order, but I'm not sure if ...
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### Characterzation of the complex numbers

There is a characterization of the real number system：complete ordered field. And the complete ordered field is unique up to isomorphism. I'm trying to charaterise the complex numbers in a similar way....
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### Non-lattice ordered fields that are not totally ordered

For the purposes of this question, totally ordered fields are ordered fields in which every element is comparable under the ordering: $x \geq y$ or $y \geq x$. Thus in this terminology, in an ordered ...
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### Different intervals in an ordered vector space over $\mathbb{Q}$

Consider an ordered vector space $V$ over $\mathbb{Q}$ in the usual language of ordered vector spaces $(<,0,-,+,\lambda\cdot)_{\lambda \in \mathbb{Q}}$. For the duration of this question, the word ...
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### Is it possible to replace hyperreal numbers with "good enough" alternatives?

The hyperreal numbers are undoubtedly interesting, generalizable, and have many nice properties, but are they really needed to solve the problems they solve? Would other, smaller fields work too? ...
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### What is the order of a function in distribution theory?

Let $\mathscr{D}(\mathcal{O})$ be the space of basic functions, $\phi$, which is equipped with convergence, and where $\phi$ vanish outside of the set $\mathcal{O}$. Let $\mathscr{D}'(\mathcal{O})$ be ...
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### Ordering of $\mathbb{R}[X]$ such that $0 < X < a$ for all positive numbers $a$

If $\mathbb{R}[X]$ denotes the field of real valued rational functions, then according to Bochnak, Coste and Roy's Real Algebraic Geometry, the unique ordering for which $X > 0$ and $X < a$ for ...
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### Real-closed fields are the existentially closed ordered fields

While reading the proof in Shorter Hodges (Thm 7.4.4) that the theory of real-closed fields has quantifier-elimination, I came across the following claim (paraphased): "Let $A$ and $B$ be real-...
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### Compatibility of formally real extensions of an ordered field

My question If $E$ is a formally real extension of an ordered field $F$, does $E$ always admit an ordering compatible with $F$? Less ambitious: What if $E$ is a formally real simple algebraic ...
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### Positive subset $\mathbb{R}_{>0}$ of $\mathbb{R}$ satisfying the order property

A field $\mathbb{F}$ is an ordered field if $\exists P\subseteq\mathbb{F}$ such that: 1. $\forall a,b\in P$, $a+b,ab\in P$. 2. Only one of the following is true $\forall a\in\mathbb{F}$, 1) $a\in P$ 2)...
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### Is there a maximal ordered field? What about $\mathbb R$?

In my real analysis class today, we revisited the fact that there is only one complete ordered field (up to isomorphism): $\mathbb R$. But this induced several thoughts I began wondering about. I ...
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### How can a field be Cauchy complete and non Archimedean

The Wikipedia page for the completeness of the Real numbers, says that “ there are non Archimedean fields that are ordered and Cauchy complete.” However, in many other places, I’ve read that non ...
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### Cardinality of integer parts of real closed fields

Every real closed field $R$ has in integer part $I$. That is, $I$ is a discrete ordered subring of $R$ such that for each $x \in R$ there is $z \in I$ such that $z \leq x < z + 1$. If $R$ is ...
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### Product of infinitesimal by inverse of non infinitesimal is infinitesimal

This is probably very simple, though I am getting confused because of my inexperience with infinitesimals. Let $K$ be an ordered real-closed field. We say that $a \in K$ is finite if it's absolute ...
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### Convex subring is local

Let $K$ be an ordered field, $R\subseteq K$ a convex subring - that is, for all $a \in K$, $a \in R$ if $x \leq a \leq y$ for some $x,y \in R$. Define $I = \{ x \in R: x^{-1} \notin R\}$. It is clear ...
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### Can the hyperreal numbers have a property akin to completeness by considering hypernatural sequences?

I’ve seen that $\mathbb{R}^*$ isn’t complete as many Cauchy sequences won’t converge, and that includes power series. In other stack exchange posts, I’ve seen that even the exponential function won’t ...
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### Whether some positive sequence approaches 0 in any ordered field

Consider an ordered field $(\mathbb{F}, +, \times, <)$ with additive identity $0$ and multiplicative identity $1$. It is not hard to show that $1\ne 0$ and then $1=1\times 1>0$. I am aware that ...
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### Can the surreal numbers be completed to form an ordered field?

The surreal number line isn’t Cauchy complete as it’s filled with “gaps”. When constructing the real numbers from the rationals, one could take the ring of all Cauchy sequences and take the quotient ...
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### Can the hyper hyper real numbers be constructed?

The hyperreal numbers can be constructed as $\mathbb{R}^{\mathbb{N}}/U$ given some ultra filter and this allows first-order statements to be transferred over to $\mathbb{R}^*$. Can this be done again ...
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### Let $(F,+,\cdot,0_F,1_F,\le)$ be an ordered field. If the set $\{x\in F: 0_F\le x\le 1_F\}$ is complete, is the whole field complete?

Let $(F,+,\cdot,0_F,1_F,\le)$ be an ordered field. If the set $\{x\in F: 0_F\le x\le 1_F\}$ is complete in the sense that the least upper bound property is satisfied, is the whole field complete, in ...
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### Question about axioms of an ordered field.

I’m currently studying Introduction to Analysis by Ross. I wanted to ask if the either - or is an inclusive or exclusive disjunction in property O1 below. I believe this should just expressing the ...
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### $F(\sqrt d)$ is an ordered field

I'm trying to solve this exercise: (hartshorne Euclid and beyond ex 15.3) Let $F$ be an ordered field, let $d>0$, and suppose that $d$ does not have a square root in $F$. Let $F(\sqrt d)$ denote ...
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### Is this way of writing the Field and Order Axioms correct?

I am currently self-studying math and am trying to write a Mathematics Cheat Sheet document that among other things includes the Field and Order Axioms. I want to write them as succinct as possible (...
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One way to define an ordered field is as a field $F$ with a relation $<$ that satisfies: For all $x,y \in F$, exactly one of $x<y$, $x=y$, $y<x$ holds. For all $x,y,z \in F$, if $x<y$ and ...