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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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 ...
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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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165 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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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 ...
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Calculus in ordered fields

Is there any ordered field smaller that the set of real numbers in which we can do calculus, also with many restrictions ? If not why ?
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Is this an ordered field?

My book says yes, but I'm not so convinced. There is: $$ F:=\{a+b\sqrt2: a,b \in \mathbb Q\} $$ Sum and product are defined in $F$ as follow. $$ (a+b\sqrt2)+(c+d\sqrt2) := a+c+(b+d)\sqrt2\\ ...
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Boundedness in real analysis

I am struggling to come up with this book example question in my real analysis textbook: Give an example of an ordered field in which {1,1+1,1+1+1,...} is bounded. If the set of natural numbers is ...
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Prove the following set has a multiplicative inverse

Let $F = \{x + y\sqrt{7} : x, y \in Q\}$ with the usual addition and multiplication operations. Show that $F$ has a multiplicative inverse. So far, I have $$\left(x_1 + y_1\sqrt{7}\right) ...
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Ordered field consequence proof [duplicate]

Hey can anyone help me get started on this proof? Prove OFC7: for elements x,y in an ordered field, (-x)(-y) = x*y. I am allowed to use the following axioms: F1. x+(y+z) = (x+y)+z; F2. x+y=y+x; F3. ...
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Ordering the field of real rational functions

Perusing the Wikipedia article on ordered fields, I encountered an interesting statement, a variation of which I am now trying to prove. Basically, I am trying to show that the set of real rational ...
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Is this proof concerning field axioms valid?

If $r$ is a rational number and $x$ irrational, prove $r + x$ and $rx$ are irrational. First, suppose $$\exists p, q \in \Bbb N : {p \over q} = r + x$$ This is equivalent to $${p \over q} - ...
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Can the ceiling function be used to prove the Archimedean property?

Recall the following definition of the Archimedean property: For each $x \in \Bbb{R}$, there exists some $n \in \Bbb{N}$ such that $n>x$. My textbook proves this by invoking the completeness ...
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How to teach a High school student that complex numbers cannot be totally ordered?

I once again need your precious knowledge! I am not sure which is the best pedagogic way to teach a High school student about why complex numbers cannot be totally ordered. When I was in High school ...
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Instead of axiomatizing ordered fields, can we axiomatize just the right half?

Since ordered fields can always be split into two halves whose only common element is $0$, namely $(\leftarrow,0]$ and $[0,\rightarrow),$ I was wondering if we can axiomatize just the right half. Note ...
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Non-archimedean matrix fields

Are there any examples of sets $A\subseteq{\Bbb C}^{n\times n}$ for which you can find operations so that $(A,+,\,\cdot\ ,<)$ is a non-archimedean ordered field? I feel like the answer is probably ...
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Help with proof concerning ordered fields and smallest elements

I have been trying to prove that every ordered field has no smallest positive element for a while now, and I think I have it worked out. However, I feel like something is missing in my proof. Here is ...
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How to show that $\mathbb{Q}_p$ cannot be ordered?

I've seen many references on this site and others to the fact that the $p$-adic numbers cannot be ordered, but the closest I've seen to a proof of this is Wikipedia's vague reference to ...
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How do extension fields implement $>, <$ comparisons?

I'm taking an abstract algebra course, and we just hit extension fields - for example, you define $\sqrt{2}$ by starting with the field $\mathbb{Q}$ and defining $\sqrt{2}$ as a solution to the ...
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The homomorphism between the real set and an complete ordered field.

I'm having troubles to prove that $f(x+y)=f(x)+f(y)$ Let $K$ be a complete ordered field and $0'$ and $1'$ be the zero and the unit of $K$. For each $n\in \mathbb N$, we have ...
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Embedding ordered number fields into $\Bbb R$

Consider a number field $K$ equipped with a total order $\le$. Is there always a field homomorphism $\phi:K \rightarrow \Bbb R$ which respects the total order, i. e. with $\phi(x)>\phi(y)$ for ...
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A field that is an ordered field in two distinct ways

Question: Explain the construction below (taken directly from Counter Examples in Analysis): An ordered field is a field $F$ that contains a subset $P$ such that $P$ is closed with respect ...
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isomorphism of Dedekind complete ordered fields

In the last chapter of Spivak's Calculus, there is a proof that complete ordered fields are unique up to isomorphism. I find the first steps in it somewhat suspicious. Specifically, I believe he is ...
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Positive cone definition for ordered rings

There are two definitions of an ordered field: you can define it as a field with a total ordering and certain axioms relating the ordering to the field operations, or as a field with a "positive ...
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Example of algebraic field not honoring the Total Order relation?

I've started studying Calculus. I've stumbled upon the definition of an 'Ordered field'. One of the requirements are for the field to honor the Total Order relation. Meaning that for every a,b, a ...
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Definable orders

Let $(K, <)$ be an order field, can I define the order "<" in $K$ ? I know that $K \models 0<a \;$ if and only if there is $b$ in the real closure of $K$ such that $b*b = a$. Can I ...
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Problem understanding a proof about powers in ordered fields

I am reading through a textbook on Analysis and have come across a question that I can't seem to make any headway with. A proof is outlined, but I can't make any sense out of it. The problem is as ...
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Non-Archimedean Fields

Are there non-Archimedean fields without associated valuation or being a non-archimedan field implies it is a valuation field? I understand that a non-Archimedean field is a field which does not ...
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Proof of Non-Ordering of Complex Field

Let $\mathcal F$ be a field. Suppose that there is a set $P \subset \mathcal F$ which satisfies the following properties: For each $x \in \mathcal F$, exactly one of the following statements holds: ...
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Archimedean ordered fields and valuations (exercise)

Definitions: Let $(K,\leq)$ be a totally ordered field and $(G,\leq)$ a totally ordered abelian group (written additively). If we denote $\mathbb{Z}a\!=\!\{na;\, n\!\in\!\mathbb{Z}\}$, then $G$ is ...
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Complete ordered field

I'm trying to prove that; If any Cauchy sequence is convergent in an ordered field F, every nonempty subset of F that has an upperbound has a sup in F. Let A be a nonempty subset of F that is not a ...
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Can an ordered field be finite?

I came across this question in a calculus book. Is it possible to prove that an ordered field must be infinite? Also - does this mean that there is only one such field? Thanks
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Ordered field $F$

If $F$ is an ordered field and $a\in F$ and $b\in F$, then how can I show that $a-b\in F$ ? $F$ has a nonempty subset $P$ such that; $a,b\in P$ ⇒ $a+b,ab\in P$ $F=P \cup \{0 \} \cup -P$ ...
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combinatorics: number of options to set a (a,b) ordered pair under terms

We have to find the number of options for setting pair $(a,b)$ under the terms: $a ⊆ b ⊆\{1, 2,\ldots, n\}$ Means, they are both subsets of $\{1, 2,\ldots, n\}$ and $a⊆ b$. I was thinking to handle ...
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An ordered field has no smallest positive element

Prove that every ordered field has no smallest positive element.
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203 views

algebraic closure and real closure are closure operators?

Are the algebraic closure (of a field) and the real closure (of a totally ordered field) closure operators (when restricted to appropriate sets of fields, so that they are maps on a set instead of a ...
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In which ordered fields does absolute convergence imply convergence?

In the process of touching up some notes on infinite series, I came across the following "result": Theorem: For an ordered field $(F,<)$, the following are equivalent: (i) Every Cauchy ...