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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Ordered fields with countable cofinality

Let $(k,+,.,0,1,<_k)$ be an ordered field, $| . |_k = x \mapsto \max(x,-x)$. If $\lambda$ is an ordinal, you can define convergent $\lambda$-sequences: maps $f: \lambda \rightarrow k$ satisfying ...
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Ordered field of rationals axiomatizable

Is there a set of sentences in the language of ordered fields whose models are precisely the rationals and any ordered field isomorphic to them?
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Does the inverse function theorem hold over $\mathbb{Q}$?

Let $f:\mathbb{Q}^n \longrightarrow \mathbb{Q}^n$. We can define what it means for such $f$ to be differentiable: (The differential will be a linear transformation $\mathbb{Q}^n \longrightarrow ...
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What is the degree of a real closure of an ordered field?

Given a subfield $F$ of $\Bbb R$, let the real-closure of $F$ be the smallest subfield of $\Bbb R$ which is real-closed and extends $F$. Since the theory of real-closed fields is complete, this is in ...
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Is every homeomorphism of $\mathbb{Q}$ monotone?

It is well known that every continuous injective map $\mathbb{R}\rightarrow\mathbb{R}$ is monotone. This statement is false for maps $\mathbb{Q}\rightarrow\mathbb{Q}$. (That is becaus $\mathbb{Q}$ is ...
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Ordering rational functions in a field

This is from Fraleigh, A First Course in Abstract Algebra, 7th edition, Section 25, "Ordered Rings and Fields", questions 10, 11,12, and 13. Here's the question: List the given elements in an order ...
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Nested Interval Property and Ordered Fields

I have been given the following as a general version of the Nested Interval Property, which is a predicate whose free-variable F ranges through the universe of all ordered fields : Every ...
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Totally ordered non archimedian fields.

While thinking about properties of real closed fields, I came across the following contradiction: Let $(k,+,.,0,1,\leq)$ be a non archimedian totally ordered field. Let us assume that $(k,\leq)$ is ...
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Show using ordering axioms that $x^2 < y^2$ for $x, y \in \mathbb{Q}$, with $0 < x < y$

I have an exercise in my last assignment of calculus: Show using ordering axioms that $$x^2 < y^2$$ for $x, y \in > \mathbb{Q}$, with $0 < x < y$ This is my solution: We have that ...
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Property of absolute value in the real numbers

To prove that $ \lvert a-b \rvert \le c-d $ for $ a,b,c,d $ in the real numbers, what needs to be shown? Is the fact that $a-b\le c-d$ enough? Or is there something more that needs to be shown?
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What does compatible mean?

Wikipedia defines ordered fields in the following way: An ordered field is a field together with a total ordering of its elements that is compatible with the field operations, i.e.: A field ...
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Ordered set and ordered field

To be clear, let me use "$=$" to mean the same element in a set, and "$\sim$" to mean neither "$>$" nor "$<$" in an order. In an ordered set even if $x\ne y$, we can still set $x\sim y$, right? ...
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Can an ordered field contain complex numbers?

I read a question about ordering of complex numbers, and saw an answer showing that there cannot exist an ordering of the complex numbers because regardless of how $i$ is placed in that order, it ...
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Are there fields and ordered fields of every infinite cardinality?

On the top of my head, I cannot think of any fields of cardinality more than that of the reals. (It is known that the process of algebraic closure does not increase the cardinality of an infinite ...
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Algorithm for Converting Rational Into Surreal Number

I'm writing a library in Haskell that represents the class of surreal numbers, which like many things can be read about here. I've run into a problem in converting between other classes of numbers ...
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Metrics (Distances) on $\mathbb{F}$ Theorem Proof

I had a question regarding a Theorem I had come across that described metrics (distances) on ordered field $\mathbb{F}.$ Here it is: Theorem: If $\mathbb{F}$ is an ordered field, then $d(x,y)=|x-y|$ ...
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Ordered Field: $|x|\le y$ iff $-y\le x\le y$

I had a question regarding this part of a theorem that describes the inequalities of the absolute value function for order field $\mathbb{F}.$ Here is the theorem: Theorem: Let $\mathbb{F}$ be an ...
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Ordered Field $\mathbb{F}$ Corollary Proof

I wanted to check my proof for a corollary on ordered fields $\mathbb{F}$. Here is the corollary: Corollary: Let $\mathbb{F}$ be an ordered field and $a\in\mathbb{F}.$ If $a>0$, then ...
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Question about supremum $\implies$ infimum

For all the proofs I've seen that if a set has the least upper bound property, then that set also has the greatest lower bound property, they assert something like if a set has lower bounds, then the ...
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Axioms for ordered fields

If i have a set of numbers, that contains every positive number, including 0 , except the negative ones, can i claim it as an ordered field? I would say no, because you can't find an $y$ for any $x$ ...
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Showing a function is a bijection without a specific function

Let $F$ be an ordered field with identities $0$ and $1'$, and define $f: \mathbb{N} \to F$ by: $f(1) = 1'$, $f(x + 1) = f(x) + 1'$ (the addition for the right hand-side is addition in the field). Let ...
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Prove $f(a + b) = f(a) + f(b)$ in an ordered field

Let $F$ be an ordered field with identities $0$ and $1'$, and define $f: \mathbb{N} \to F$ by: $f(1) = 1'$, $f(x + 1) = f(x) + 1'$ (the addition for the right hand-side is addition in the field). So ...
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Estimations in a ordered field?

My Problem: I am stuck with a proof strategy on the following: So i have got an ordered field $ (K,+,*,<) $ given. I also have $x,y\in K$ and $0\le y < x$ I have to proof that, for every n ...
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A field between $\mathbb{Q}$ and $\mathbb{R}$ ?

I really have trouble understanding a task. We've got $p\in$ P, while P are all prime numbers. Now we construct a field $$\mathbb{Q}[\sqrt{p}]:=\{x+y\sqrt{p}:x,y \in \mathbb{Q}\}$$ The Task is to ...
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Ordered field proof

Prove that in any ordered field F, $a^2+1>0$ for all $a\in \!\,F$. There are 2 cases, Suppose $x\ne 0$. Then $x^2>0$. Suppose $x=0$. Then $x^2=0$. In both cases, $x^2+1>x^2\ge\ 0$ Does ...
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Absolute value proof

Prove that $|xy|=|x||y|$ hint: consider the 4 cases, x and y both positive, both negative, x is positive while y is negative, and y is positive while x is negative. I am not really sure how to go ...
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How to prove that this enough to determine the structure of $\mathbb R$?

Let $F$ be a field and let $P\subset F$ such that the sets $P$, $\left\{ 0\right\} $ and $-P:=\left\{ -p:p\in P\right\} $ form a partition of $F$ and $x,y\in P\Rightarrow x+y\in P\wedge xy\in P$. I ...
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Show that if a subset of an ordered field has a least upper bound, then any two least upper bounds for it have to coincide

Show that if a subset of an ordered field has a least upper bound, then any two least upper bounds for it have to coincide Should I prove by contradiction then turn my proof into a direct proof using ...
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Can an orderable field always be ordered in a way that extends a given subfield's ordering?

Let $F$ be an orderable field, and let $\:\langle E,\hspace{-0.03 in}\leq \rangle \:$ be an ordered subfield of $F$. Does it follow that $F$ can be made into an ordered field in a way that extends ...
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How to prove a finite field is not ordered?

I have a set S={0,1}, and the addition and multiplication rules are \begin{array}{c|cc} +&0&1\\ \hline 0&0&1\\ 1&1&0 \end{array} \begin{array}{c|cc} *&0&1\\ \hline ...
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Does the mean of two positive numbers obey these basic inequalities?

Suppose $a,b$ are positive real numbers. Does it follow that $$ a < \frac{a+b}{2} < b $$ provided $a < b$?
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How to prove: If $x \geq 0 $ and $x \leq \epsilon$, for all $\epsilon > 0$, then $x = 0$?

I am trying to prove this problem for my homework. I am having some difficulty with this, because we are just supposed to use several ordered field axioms, the four order axioms, and several basic ...
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Is$\ +\infty$ greater than any other number (surreal, superreal, hyperreal, …)?

Let$\ \mathbb{A}$ be an arbitrary totally ordered set and consider the largest element of the set of extended real numbers,$\ +\infty$. Can we say that$\ +\infty > \chi $, for *any*$\ \chi \in ...
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Proving Properties in Ordered Fields

Refer to Definition 1.3, which states, an ordered field is a field F that is ordered set with the following additional properties: ...
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Is 1/2 less than 1?

Prove that $\frac 12 \lt 1$ We're in the reals and we have the Field Axioms and Order Axiom. I know that for $x \lt y$, $y - x \gt 0$. However I think I'd be assuming what I;m trying to prove if I ...
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Prove that if $F$ is an ordered field in which every non empty set which has an upper bound also has a supremum , then $F$ is archimedean

Prove that if $F$ is an ordered field in which every non empty set which has an upper bound also has a supremum , then $F$ is archimedean Attempt: If $F$ is an ordered field, then it possesses a ...
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If $y,z$ are elements of an archimedean field $F$ and if $y<z$, then there is a rational element $r$ of $F$ such that $y<r<z$

If $y,z$ are elements of an archimedean field $F$ and if $y<z$, then there is a rational element $r$ of $F$ such that $y<r<z$ The proof begins with saying that it is no loss of generality ...
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In an Archimedian Field $F$, there is a positive rational element $r$ such that $r < z$ for any $ z>0$ in $F$

In an Archimedian Field $F$, there is a positive rational element $r$ such that $r < z$ for any $ z>0$ in $F$ . Is this statement true? Attempt: An archimedian field $F$ is an ordered field in ...
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Replacing the “if $x ≤ y$, then $x + z ≤ y + z$” axiom in Reals.

How can I prove that we cannot (or maybe can) replace preservation of order under addition i.e. "If $x \leq y$, then $x + z \leq y + z$ with "if $0<x$ and $0<y$ , then $0<x+y$" in axioms ...
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Are order isomorphic real closed fields isomorphic?

There are counterexamples to order isomorphisms of ordered fields being field isomorphisms, see Is the multiplicative structure of a totally ordered field unique?. However, Wikipedia suggests that for ...
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Proof that all ordered fields are in the Surreals

It says on Wikipedia that any ordered field can be embedded in the Surreal number system. Is this true? How is it done, or if it is unknown (or unknowable) what is the proof that an embedding exists ...
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Ordered field with nested intervals axiom without Archimedean axiom

Does an ordered field $F$ satisfying the Cauchy-Cantor axiom but not the Archimedean axiom exist? Cauchy-Cantor axiom: Every system of nested intervals $I_1 \supset I_2 \supset \cdots \supset ...
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Existence and uniqueness up to isomorphism of the real numbers from axioms

Pretty much what the title says: how does one prove the existence and uniqueness of the real number system from the ordered field axioms together with the least-upper-bound property (or maybe some ...
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Models of ZFC as Ordered Fields

Question: Let $M$ be a set such that $\langle M,\in\rangle\vDash ZFC$. Consider the partial order $\langle M,\subseteq\rangle$. Using order extension principle there is a linear order $\subseteq^{*}$ ...
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Exercise 5 .39 page 46 Real and Abstract Analysis [Hewitt and Stromberg]

How to show that every non-Archimedian field contains a subfield algebraically and order isomorphic to $\mathbb{Q}(t)$, where $\mathbb{Q}(t)$ is the field of rational functions with coefficient in ...
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Density of rationals in an ordered field

Given an ordered field $F$, is it equivalent to say that $\mathbb{Q}$ is dense in $F$ with respect to the order topology and to say that $\forall x\in F,\forall y\in F,x<y\Rightarrow\exists q\in ...
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Unicity of order/positives in ordered field

I was remembering some stuff from my calculus course, and I got to the following question: By a field order, I mean a total order $\leq$ on a field $\mathbb{F}$ which is invariant by addition and by ...
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Multiplicative Inverse all between 0 and 1

Why do all "the" mutliplicative inverses in an ordered field stick in a bunch between 0 and 1? It's given that $x>y>0 \Rightarrow 0<x^{-1}<y^{-1}$ and the 1 is its own inverse. So you get ...
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Are the multiplicative inverses contained in the cone?

Definition: A cone of a field $F$ is a subset $P$ of $F$ such that : $x,y\in P\implies x+y\in P$ $x,y\in P\implies xy\in P$ $x\in F\implies x^2\in P$ I would like to know if the third condition is ...
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Ordered field of Polynomials.

List the following functions from smallest to largest. $$\frac{x^2 + 2}{x-1},\frac{x^2 - 2}{x+1},\frac{x + 1}{x^2-2},\frac{x + 2}{x^2-1}$$ To solve the problem, I believe we compare each fraction ...