Questions tagged [ordered-fields]
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.
445 questions
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Is the the nested interval property equivalent to Cauchy criterion without Archimedean property?
Recall the following properties of the reals.
Nested interval property. For every sequence $I_1, I_2, I_3,\ldots$ of closed and bounded intervals satisfying
$$I_1 \supseteq I_2 \supseteq I_3\supseteq \...
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Problem with Proof Reals Are Essentially Unique
I am trying to understand a proof of the fact the real numbers form the only ordered field having the least upper bound property.
https://web.williams.edu/Mathematics/lg5/350/RUnique.pdf
The proof is ...
user1705636
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Necessity of proving addition to both sides of an equation (axiomatic construction of Q)
I am beginning my first course in real analysis. We have begun with an axiomatic construction of the rational numbers. I've included our particular set of axioms at the end for reference.
My professor ...
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Whether Cauchy complete and Cantor complete are equivalent in terms of ordered field
For reference, "Cantor complete" means that every nested sequence of bounded closed intervals has non-empty intersection.
It is easy to show that the conditions "Cauchy complete" ...
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Textbooks combining order theory, topology and algebraic structure
It looks like a greedy title.However, when I learn about construction of real numbers and different equivalent ways to do completion like Dedekind completion and Cauchy completion from some papers ...
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Which fields have unique ordering?
$\Bbb{R}$ has a unique ordering, as does $\Bbb{Q}$, and all real-closed fields. Furthermore, $\Bbb{Q}(\sqrt[3]{2})$ has a unique ordering, but this isn't trivial to see, and many other algebraic ...
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How can countable left-orderable groups be densely left-ordered?
While trying to understand a proof of the fact that a countable left-orderable group (a countable group with strict total ordering that is invariant under left group multiplication) acts on $\mathbb{R}...
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Given a real closed field $R$, can there be automorphisms of $R$ other than the identity?
Given a real closed field $R$, can there be automorphisms (order preserving field automorphisms) of $R$ other than the identity?
For specific choices of $R$ there can only exist the identity, for ...
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Generalization of formally real fields to fields equipped with an involution
Let $K$ be a field. If $0$ cannot be written as a sum of nonzero squares in $K$, then $K$ is a formally real field.
Now, if $K$ is a field with involution $*$, do we have a name for fields in which $0$...
- 2,611
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Do different orders on the same ordered field coincide up to isomorphism?
Ordered field means the field $F$ equipped with a totally order $>$, which satisfies
· $\forall x,y,z \in F, y > z \Longrightarrow x+y > x+z$;
· $\forall x,y \in F, x,y > 0 \Longrightarrow ...
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A maximal totally real subfield of a quadratically closed field with characteristic $0$ is of index $2$
Call a field formally real if there exists an order on it making it an ordered field. Let $L$ be a field with characteristic $0$. As discussed in this post, there exists a subfield $K$ which is ...
- 2,783
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Analogue of Bolzano–Weierstrass principle for ordered sets
Is it true that for a linearly ordered set $A$, completeness is equivalent to the fact that for any infinite subset $B$ of an interval (interval is the set of $x\in A$ such that $a \leq x \leq b$, for ...
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Showing that each natural, $x$, of an ordered field can be uniquely represented in the form: $x= n\cdot 1_F$, where $n$ is a natural of $\mathbb{R}$
Problem statement:
Show by induction that each natural element $x$ of an ordered field $F$ can be uniquely represented as $x=n\cdot 1_F$, where $n$ is the natural number in $\mathbb{N}_{\mathbb{R}}$ ...
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Does field isomorphism preserve order relation?
Let $(F,+_\mathrm{f},\cdot_\mathrm{f},<_\mathrm{f})$ and $(G,+_\mathrm{g},\cdot_\mathrm{g},<_\mathrm{g})$ be two ordered fields. Let $f:F\to G$ be a field isomorphism. (So, $f$ is bijective, and ...
user1345781
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Permutations and partial orders
Consider the set of all permutations of $n$ numbers $\mathfrak{S}(n)$. Each permutation can be seen as a total ordering relation of $n$ elements $a_1,...,a_n$, such that
\begin{equation}
a_{\pi(1)}<...
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Characterzation of the complex numbers
There is a characterization of the real number system：complete ordered field. And the complete ordered field is unique up to isomorphism. I'm trying to charaterise the complex numbers in a similar way....
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Is it possible to replace hyperreal numbers with "good enough" alternatives?
The hyperreal numbers are undoubtedly interesting, generalizable, and have many nice properties, but are they really needed to solve the problems they solve? Would other, smaller fields work too?
...
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Ordering of $\mathbb{R}[X]$ such that $0 < X < a$ for all positive numbers $a$
If $\mathbb{R}[X]$ denotes the field of real valued rational functions, then according to Bochnak, Coste and Roy's Real Algebraic Geometry, the unique ordering for which $X > 0$ and $X < a$ for ...
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Compatibility of formally real extensions of an ordered field
My question
If $E$ is a formally real extension of an ordered field $F$, does $E$ always admit an ordering compatible with $F$?
Less ambitious: What if $E$ is a formally real simple algebraic ...
- 3,352
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Adjoining greatest and smallest elements to uncountable ordered field
As titled. With $\mathbb{R}$, we can adjoin $+\infty$ and $-\infty$ as the greatest and smallest elements and define $\overline{\mathbb{R}}:=[-\infty, +\infty]$. But if we have an ordered field $\...
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Proof of uniqueness of real-closure of an ordered field
I'm reading the proof of uniqueness of real-closure of an ordered field $F$, that is an algebraic extension $R$ of $F$ such that $R$ is real-closed and the unique order on $R$ extends that of $F$. I ...
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Strong homogeneity of order type of real closed fields
In Keisler's Infinitary Logic Book, he notes at one point that if we take a real closed field of cardinality $\kappa$, the ordered set is countably strongly homogeneous, in the sense we can find an ...
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Does every real-closed field satisfy the first-order least upper bound axiom schema?
The theory of real-closed fields is usually axiomatized by the axioms of ordered fields, along with an axiom for existence of square roots of positive elements, and also an axiom schema stating that ...
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First derivative test for polynomials over ordered fields
I am interested in the following generalization of the first derivative test for real functions: Let $K$ be an ordered field and $p\in K[X]$ a polynomial. Consider an interval $I\subseteq K$ on which ...
- 2,411
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Visualize the Completion of (the Ordered Field) of Rational Functions
Every ordered abelian group $G$ can be completed to give a larger ordered abelian group $\bar{G}$. The original abelian group $G$ embeds into $\bar{G}$ as a dense subset, and every non-empty subset of ...
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