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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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 $x ā‰¤ y$, then $x + z ā‰¤ y + z$” 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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Ordered field - Adding inequalities (Looking for a proof)

In an ordered field, it is allowed to add two inequalities. Where can I find a detailed proof?
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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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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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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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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 $f:Nā†’\{1,1+1,1+1+1,...\}$ 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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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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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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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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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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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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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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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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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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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: ...