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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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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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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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 ...
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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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“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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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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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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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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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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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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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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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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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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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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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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Real Analysis-greatest lower bound/infimum

Suppose that S is a complete ordered set as defined in Definition 1.4, part vi (see below). Show that this implies the greatest lower bound property for S, i.e., show that if A is any subset of S ...
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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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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 $f<q_1$, ...
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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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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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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 ...