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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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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3answers
352 views

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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1answer
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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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1answer
47 views

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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1answer
56 views

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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23 views

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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1answer
20 views

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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1answer
39 views

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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32 views

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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Order of analytic functions f(z) and g(z)

If $n=ord_{z_0} (f)$ and $m=ord_{z_0}(g),$ suppose that $n < m.$ Show that $ord_{z_0} (f+g) = \min(n,m).$ Also give an example where n=m but $ord_{z_0} (f+g) > \min(n,m)$ Hints on how to ...
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44 views

Is there anyone who could help me with this problems? [closed]

Let $R=\mathbb{Z}[\sqrt7]=\{ a+b\sqrt7 ~| a,b\in \mathbb{Z}\} $ define addition and multiplication by $(a_1+b_1\sqrt7)+(a_2+b_2\sqrt7)=(a_1+b_2)+(b_1+b_2)\sqrt7$ ...
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1answer
60 views

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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1answer
19 views

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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31 views

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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1answer
22 views

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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1answer
21 views

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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1answer
43 views

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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2answers
38 views

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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1answer
61 views

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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25 views

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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1answer
29 views

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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2answers
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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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1answer
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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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71 views

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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138 views

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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34 views

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 ...
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Interesting question with ordering integers

The numbers $1,2,3,...,101$ are written down in a row in some order. Is it always possible to cross out $90$ numbers in a way such that all $11$ numbers left will stay either in increasing or in ...
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Is there an accepted definition of the “coefficients” of a multivariate rational function?

I'm reading Complexity and Real Computation by Blum, Cucker, Shub, and Smale. In in defining "machine constants" of a Blum-Shub-Smale machine, they talk about the "coefficient" of a multivariate ...
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1answer
434 views

Every ordered field has a subfield isomorphic to $\mathbb Q$?

I'm going through the first chapter in a text on real analysis, which contains preliminaries on ordered fields, the real numbers, etc. Supposedly I had learned about such things already, in calculus, ...
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Correspondence between an order of rational function field and a Dedekind cut

In the first page of Real Algebraic Geometry by Jacek Bochnak, Michel Coste, Marie-Francoise Roy, they briefly connect an order of the rational function field $R(X)$ with a Dedekind cut ...
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228 views

Is the multiplicative structure of a totally ordered field unique?

Is it possible to find totally ordered fields $K$ and $L$, and a map $f: K \to L$ that is an isomorphism of the ordered additive group structures such that $f(1)=1$, but which is not an isomorphism of ...
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A finite field cannot be an ordered field.

I am reading baby Rudin and it says all ordered fields with supremum property are isomorphic to $\mathbb R$. Since all ordered finite fields would have supremum property that must mean none exist. ...
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Splitting field of irreducible polynomial $f(x)=x^4 +ax^2+b$ over field with charactersitic not equal to 2

this is from Seth Warner's Classical Modern Algebra: Let $K$ be a field whose characteristic is not $2$, let $f=x^4+ax^2+b$ be an irreducible polynomial over $K$, and let $L$ be a splitting field of ...
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Why this map preserves order?

How to prove that $f:N→\{1,1+1,1+1+1,...\}$ where 1 is an identity element of ordered field, is order-preserving? I guess that maybe property if $a < b$ then $a + c < b + c$ can be useful, but ...
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Question about polynomials of odd degree with no zeros in formally real fields which are maximal to the property of being ordered

I have encountered this argument while reading Tent and Ziegler's "Course in model theory", and I don't know why it is justified. It arises during the proof that every ordered field has a real ...
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Banach spaces over complete fields with their own absolute value

Let $F\hspace{.03 in}$ be a field, and let $E\hspace{.03 in}$ be an ordered subfield of $F$. Let $\;\; |\hspace{-0.03 in}\cdot\hspace{-0.03 in}| \: : \: F \: \to \: E \;\;$ be such that for all ...