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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Do analytic properties hold in an arbitrary ordered field?

Given an ordered field, we can view it as a field formally equipped with some analytical notions which come from $\mathbb R$, like order and derivative. So I'm curious if $F$ also carries some ...
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Rules of calculus for non-complete ordered fields

Suppose that $K$ is a totally ordered field. We can express the properties of continuity and differentiability just as in the real numbers. Now suppose we have a function $f: K \to K$. I was wondering ...
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Completing ordered Fields

How do these two forms of completion behave (in NBG) when fields are authorized to be proper classes? $(i)$: Every ordered field has a real closed, algebraic extension. $(ii$): Every ordered field ...
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Algorithm for ordering on an algebraic number field

Given an algebraic field extension of the rationals $Q(P(X))$, where $P(X)$ is a polynomial in $X$, how do I algorithmically define an ordering on $Q(P(X))$ that is compatible with a specific real ...
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Does every non-Archimedean ordered field contain $\mathbb Q (x)$ as an ordered subfield?

Every ordered field $\mathbb F$ contains $\mathbb Q$ in a canonical way. If the field is not Archimedean there exists an $x>n$ for all $n \in \mathbb N$. Since we are dealing with a field any ...
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What is the correct way of adjoining an “infinite” point to a totally-ordered field, such that the end result is still totally-ordered?

Let $Q$ denote a totally-ordered field. Now adjoin an infinite element $\infty$ to $Q$, by equipping the field of rational functions $Q(\infty)$ in the indeterminate $\infty$ with the least preorder $\...
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Partial Order Relations with irreflexive definitions

Define a relation R on the set of real numbers by (x,y) R if and only if x - y = 0. Determine if the relation R is a partial order. If it is not a partial order, explain which property or properties ...
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Dedekind cuts in Archimedean field

Let $F$ be an ordered field with the property that for all Dedekind cuts $(A,B)$ and all $n\in \mathbb{N}$ there are $a\in A$ and $b\in B$ such that $b-a< \tfrac{1}{n}$. I would like to conclude ...
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28 views

Why is the field of rational functions not dedekind complete?

The field of rational functions $F_{\mathbb{Ra}}$ in one variable is a classic example of a non-archimidean ordered field. For an ordered field to be non-archimidean, it must not be dedekind complete ...
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Close points with bounded minimal polynomial

Let $F$ be an ordered field, let $F'$ denote its real closure. Let $d$ be a natural number and let $A$ be a bounded subset of $F'$, all of whose elements are of degree $d$ over $F$. Is there a map $f:...
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154 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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Formally real field with two different orders

If $F$ is a formally real field, then there exists a total order relation on $F$ which is compatible with its sum and product, but it needs not be unique. a) How different can be two (compatible with ...
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27 views

Imposing ordering on $Q(\mathbb{R}[x])$

I'm working on a question that asks to show that the field of fractions of the polynomial ring with real coefficients is an ordered field by defining $$\frac{p}{q} > 0 \iff \frac{a_m}{b_n} > 0 $$...
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1answer
31 views

Proof of $ \forall a \in \Bbb{R}: -a = (-1) a $.

Problem. Prove that $ \forall a \in \Bbb{R}: -a = (-1) a $. I already have a proof but I would like to see another. :) Proof by contradiction Suppose that $ -a \neq (-1) a $. Then \begin{align} -...
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Completion of surreal subfields

Let $\kappa$ be a regular uncountable ordinal. Let $No(\kappa)$ denote the field of surreal numbers of birthdate $ < \kappa$. In Fields of surreal numbers and exponentiation (2000), P. Ehrlich and ...
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550 views

“Ordering” of Complex Plane

I have heard that the Complex Numbers do not form an ordered field, so you can't say that one number is larger or smaller than another. I've always been fine with this, but I recently wondered what ...
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1answer
128 views

Example of a chain without a supremum in a non Archimedean ordered field

I give here the example of a non-Archimedean ordered field. I know that the field is not order complete. What is a simple example in that field of a chain without a supremum?
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1answer
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Show that formally real fields can be totally ordered into ordered fields

A field $F$ is said to be formally real if the only solution to $\sum_{i=1}^{n}{x_{i}^{2}}=0$ is the trivial solution $x_{i}=0$, for any $n \in \mathbb{N}$. How do I show that there exists a total ...
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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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1answer
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How to proof that a field is complete order field?

I know what an ordered field is but how to actually proof that a field, for example $Q$ (rational numbers), is an ordered field?
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1answer
299 views

Relatively tight upper and lower bounds for surreal numbers

As is well, known, the surreal numbers have gaps. As far as I understand, this means that a set of surreal numbers will not always have a supremum or infimum in the surreal numbers. So I thought ...
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1answer
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Uniqueness of Limit in an Ordered Field

Let $K$ be an ordered field. Then $K$ contains a unique copy of the ordered field $\mathbb{Q}$. Also, we can define the absolute value $|\cdot|$ and sequence convergence in exactly the same way as in $...
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Limit of the derivative and LUB

Let $(k,+,.,0,1,<)$ be an ordered field. In the folowing definitions, all numbers and notions are derived from the ordered field structure of $k$, and $a < c$ are generic elements of $k$. ...
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1answer
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Finding order of a point on eliptic curve

Just started studying eliptic curves and am having trouble with this question. An explanation/solution would be much appreciated. Find the order of the point X on the elliptic curve $E/Q$ for the ...
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Integer parts isomorphic?

Let $F$ be a real closed field. It is known$^{[1]}$ that $F$ has an integer part, that is, a subring $A$ such that $\forall x \in F, \exists ! a \in A, a \leq x < a+1$. Are all integer parts over ...
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1answer
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Why is there no 'trivial' ordering on the complex numbers

The ordering I'm trying to consider is simply for $x,y \in \mathbb{C}$ then $x=y$. I'm going through Rudin's Principles of Mathematical Analysis and the only restrictions he gives for an ordered ...
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27 views

Field property proof

I have tried quite a lot of "turn and twists" but I just can`t get around it. Here it is : if x > y then x > z > y for all $x,y$ in the partially ...
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1answer
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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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Find all factored pairs of (a,b) such that…

Determine all possible ordered pairs (a, b) such that $a − b = 1$ $2a^2 + ab − 3b^2 = 22$ I've gone as far as factoring the left side of the second equation: $(a-b)(2a + 3b) = 22$ $(...
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on the lexicographic order on $\mathbb{C}$

I know that the lexicographic order on $\mathbb{C}$ turns it into an ordered set. I just wonder if under this ordering, $\mathbb{C}$ has the least upper bound property
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Number of sulutions for $c=x^2$ in an ordered field

If $D$ is an ordered field and $c = x^2$, where $c \in D$. How many solutions are possible? Please provide a proof or an example. I know that there could be no solutions if, for example, $D \in \...
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General exponential inequality?

The theory $Th_{\exp,\mathrm{fields}}$ of exponential ordered fields is the first-order theory over $\left\langle +,\cdot,0,1,<,\exp\right\rangle$ whose axioms state that the model is an ordered ...
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Automorphism group of the class of surreal numbers

Do we know the group (Group) of automorphisms of the ordered Field of surreal numbers? I feel the different ways to see the surreal numbers should provide us with several ways to define interesting ...
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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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1answer
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Every ordered field that has the least upper bound property is isomorphic to the real number system.

Okay, so here's a theorem from Rudin: "Every ordered field that has the least upper bound property is isomorphic to the real number system." Here's a definition: "Ordered fields are isomorphic if ...
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Can a rational bigger than product of two reals be written as a product of two rationals bigger than each real?

Suppose you have $f,g \in \mathbb{R}$ with $f,g>0$ and $q \in \mathbb{Q}$ such that $fg<q$. How can I prove that there exist $q_1, q_2 \in \mathbb{Q}$ s.t. $q=q_1 q_2$ and $f<q_1$, $g<q_2$?...
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Height of an ordered field

I'm studying ordered fields, and a specific notion regarding ordered fields that I will denote here by their "height". If $k$ is an ordered field, and $\alpha$ is a non-empty ordinal, a ruler of ...
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Ordered fields are real?

This question is motivated by another question posted earlier today. Let $F$ be a field endowed with an embedding $F\hookrightarrow\Bbb R$. Then the ordering of $\Bbb R$ induces an ordering on $F$. ...
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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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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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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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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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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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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? ...