# Tagged Questions

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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### 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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### 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 \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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### “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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 ...
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? ...