# 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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### How Can You Define Successors Over The reals?

I've been going through Introduction To Modern Set Theory by Judith Roitman, and am confused by her exposition of well orderings. She gives the following proof that every element $x$ of a well-ordered ...
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### Why is $-1>0$ not enough?

Theorem: Prove that no order can be defined in the complex field that turns it into an ordered field. Proof: Suppose complex field is an ordered field. So, either $i$ or $-i$ must be positive. ...
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### Prove that arithmetic mean is always greater than geometric mean in ordered fields

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. ( More details ) I want to prove that this inequality holds ...
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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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### 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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### 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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### 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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### 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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### 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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### 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?