# 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.

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### Show that there are two total orderings of $\textbf{Q}(\sqrt{2})$ under which it is an ordered field.

Let $\textbf{Q}(\sqrt{2})$ be the set of all real numbers of the form $r + s\sqrt{2}$, with $r,s\in\textbf{Q}$. Show that $\textbf{Q}(\sqrt{2})$ is a subfield of $\textbf{R}$. Show that there are two ...
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### $b^{x+y}=b^x b^y$ for $x$, $y \in \mathbb{R}$ and $b>1$.

This is from Baby Rudin and it seems to have caused a lost of confusion and controversy among classmates and colleagues, but amidst all of this I found it quite simple and was wondering if I was ...
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### Name and definition of the standard ordering on $\mathbb{R}$

Is there any name for the standard ordering on the reals? I typically just see descriptors like "the usual ordering", "the typical ordering", or - well - "the standard ordering". But is there any ...
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### Is it true that if $p,q \in K_{>0}$, then $[0,pq) \subseteq [0,p)[0,q)$ in any ordered field?

It seems possible that the following holds: Conjecture. Let $K$ denote an ordered field. If $p,q \in K_{>0}$, then $[0,pq) \subseteq [0,p)[0,q).$ More precisely: If $p,q \in K_{>0}$,...
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I was thinking about how to define the real number system axiomatically, and can't find anywhere a proof that $$\left[\forall n \in \mathbb{N}\left(|x| \le \frac{1}{n}\right)\right] \Rightarrow [x = 0]... 1answer 22 views ### Are order-preserving field embeddings unique? Let K be an ordered field with an embedding into \mathbb R,$$f:K\hookrightarrow\mathbb R,$$where f is order preserving. Is f unique? Follow up from this question of mine (same question ... 2answers 18 views ### Total ordering on a field defined by a subset This is an exercise in my analysis class, which I only could finish partially. Let (F,+,\cdot) be a field and let \mathbb{P} \subset F be a set with the following properties: 1) 0 \in \mathbb{... 1answer 64 views ### Examining the theorem, for all \epsilon \gt 0, if x \le y + \epsilon, then x \le y For this theorem, I understand the case in which x \lt y + \epsilon. From there it follows visually that x could either be equivalent to y itself, or x could also be less than y; in either case, the ... 1answer 48 views ### Can the Dedekind completion of the hyperreal numbers be embedded in an ordered field? This answer shows that the Dedekind completion of the set of hyperreal numbers, endowed with the usual definitions of addition and multiplication of Dedekind cuts, is not an ordered field. But my ... 0answers 24 views ### How to rearrange distance matrix in order of (dis)similarity for coordinates in d-dimensions? I am trying to self-learn more about distance matrices and their application in different clustering techniques. Suppose I have a distance matrix for N coordinates (... 0answers 23 views ### Inductive sets in ordered field In a real analysis book, I read that an ordered field has at least 2 inductive sets and also can have many more. I am not able to understand how "at least 2"? Please clarify. 1answer 18 views ### Relation between multiplicative and additive identity in an ordered field In class, we prove that 1>0 by using order axioms of a field. The question is can we conclude that multiplicative identity is always less than the additive identity for any arbitrary ordered field? 2answers 54 views ### To prove (2,2) is an interval in R The definition of an interval in an ordered field is a set that always contains the entire closed interval between any two of its elements. How do we prove (2,2) is an interval which is an empty set ... 1answer 49 views ### A Sequence which is co-Cauchy with a Positive Cauchy Sequence is Positive Indeed, this question arises when one wants to prove the well-definedness of an order while constructing \mathbb{R} from equivalence classes of Cauchy sequences in \mathbb{Q}. Definition 1. A ... 2answers 51 views ### show 1/x does not equal  0 [closed] Suppose that x\neq 0, show that \frac{1}{x}\neq 0 This seems like it should be simple, but I don't even know how to start to prove this. 1answer 34 views ### A property of \gamma_1 extensions of real closed fields. For each real closed ordered field \mathbb{F}, \sum_\omega({}^m\mathbb{F},S) denotes the set of all definable subsets in {}^m\mathbb{F} with parameters in S. We consider \sum_\omega({}^m\mathbb{... 1answer 37 views ### Show that extension \mathbb{R}(x) \supseteq \mathbb{R} is a real extension (ordered extension) Problem: Denote \mathbb{R}(x) be the quotient field of polynomial ring in single variable \mathbb{R}[x]. Show that extension \mathbb{R}(x) \supseteq \mathbb{R} is a real extension (ordered ... 1answer 43 views ### Show that the real field \mathbb{R} has a unique ordering and indicates that ordering Problem: Show that the real field \mathbb{R} has a unique ordering and indicates that ordering. My question: We knew that \le is an ordering on \mathbb{R}. Do we need to prove that \le is an ... 2answers 61 views ### Maximal algebraic ordered field extensions of \mathbb{Q} The real algebraic numbers form a maximal algebraic ordered field extension of \mathbb{Q} in the sense that they are an algebraic ordered field extension of \mathbb{Q}, and no other ordered field ... 1answer 55 views ### Archimedean Property and limit of q^n for 0<q<1 How could I show, that if \lim_{n\rightarrow\infty}q^n=0 for 0<q<1 in an ordered field (not \mathbb{R}), that the ordered field is archimedean, meaning for every x,y in this field with ... 1answer 39 views ### Can a real-closed field of uncountable cofinality have a countable gap? Let \mathbb{F} be a real-closed field with \mathrm{cof}(\mathbb{F})\geq\omega_1. For subsets A and B of \mathbb{F}, we say that \langle A, B \rangle is a pregap if \forall a \in A \ \... 2answers 53 views ### Proof that if \lim_{n \to \infty } \frac{a(n+1)}{a(n)} =0 for a:\mathbb{N}\mapsto K, a(n)>0 \forall n\in \mathbb{N}, then \lim(a)=0 I want to prove: Is K an ordered field and a:\mathbb{N}\rightarrow K with a(n)>0\;\forall n\in \mathbb{N}. If \lim_{n \to \infty} \frac{a(n+1)}{a(n)} =0, then \lim a=0. Please let me ... 1answer 65 views ### Understanding why complicated argument is necessary when looking for index 2 subgroups I have question concerning the proof of Theorem 1 in On ordered skew fields. The author uses Zorns lemma to construct a maximal subgroup P \subset K^* with -1 \notin P and S \subseteq P. We know ... 0answers 26 views ### Solve the problem with numbers from 1 to 101 written in a random order Numbers from 1 to 101 are written in a row in a random order forming a list. Prove that you can remove 90 numbers from this list such that remaining numbers will be sorted in the ascending or ... 2answers 63 views ### A definition of ordered field Definition: An ordered field is field F equipped with a linear order < such that 1 > 0 For any a,b \in F,$$ a,b > 0 \quad \Longrightarrow \quad a+b,\;ab > 0.$$I like this ... 0answers 41 views ### Is \mathbb{Q} the smallest ordered field up to isomorphism? Pretty simple question. Does there exist a ordered field smaller than (i.e. is a strict subset of) \mathbb{Q}? It seems like we can't go any smaller than \mathbb{Q}. Is this true? Why? 1answer 34 views ### No single axiom stating non-Archimedeanity [duplicate] An ordered field K with the ordering < is Archimedean if for any x \in K there exists n \in \mathbb{N} (where \mathbb{N} is the copy of the natural numbers in K) such that |x|<n. ... 1answer 23 views ### Proof of c > a - 2b if |c - a| < b Let (K,+,\cdot, P) be a totally ordered field. How can one prove that for a,b,c \in K if |c - a| < b it holds that c > a - 2b? I tried it out with numbers and couldn't find a ... 1answer 17 views ### Proof of ca^2 < cb^2 in an ordered field Let (K,+,\cdot, P) be a totally ordered field. How can one prove that for a,b,c \in K it holds that if a < b and c > 0 then ca^2 < cb^2? I know that if a < b \wedge c > 0 ... 1answer 30 views ### How can I prove this statement without using reduction to absurdity? \forall a,b\in\mathbb R[\forall c\in \mathbb R(c>a\implies c>b)\implies a\ge b] 0answers 31 views ### First-order properties of Euclidean fields (instead of real closed fields) Let \mathbb{F} be an ordered field. \mathbb{F} is called Euclidean if \forall x>0 \in \mathbb{F}\ \exists y \in \mathbb{F} : y^2=x, i.e. a square root exists. \mathbb{F} is called real ... 1answer 51 views ### What does “\mathbb{R}-complete” mean in Woodin's paper? I'm reading Woodin's paper "A Discontinuous Homomorphism from C(X) without CH". In this paper, Woodin defined "\mathbb{R}-complete" as the following: DEFINITION 1. Suppose that \mathbb{H} \subset ... 1answer 46 views ### use the ordered field \mathbb{R}. For |x−y|<0.01 and x,y\in(0,2), show that |x^2−y^2|<0.04 only use properties of absoulte values [closed] I Don't know how to start this question can someone please help me? Use the ordered field \mathbb{R}. For |x−y|<0.01 and x,y\in(0,2), show that |x^2−y^2|<0.04 1answer 49 views ### Equivalence relation on ordered field Let p be a prime number. Define an equivalence relation ∼ on Z as: n ∼ m if n−m is divisible by p. For n ∈ Z, let [n] be the equivalence class of n with respect to this equivalencerelation. LetZp ={[n]... 1answer 28 views ### “Simple” ordered fields proof: show that 0<y^{-1}<1 Given that x>0 and y=x+1 show that 0<y^{-1}<1 specifying what proprieties of the Ordered Field you are using. I really struggle to understand whether I have demonstrated the theorem ... 1answer 80 views ### Galois Group of Ordered Field containing Square Roots Given a ordered field K such that every positive element 0<x has a root in K we can show that any endomorphism f:K\rightarrow K preserves the ordering of K and turns out to be \text{id}... 3answers 40 views ### Is it useful that k(X) is an ordered field if k either is? I have recently found that if k is an ordered field, k(X) is also an ordered field. Proof below, in the proof that order is defined. My question: I have been never told that a field of ... 2answers 108 views ### Prove that there exists no such total order Prove that there does not exist a total order \leq on \mathbb{C} such that (i) for all x,y,z \in \mathbb{C}, if x\leq y, then x+z\leq y + z; (ii) for all x, y \in \mathbb{C}, if x \geq ... 2answers 70 views ### The Order of sets like \mathbb Q and Justification of the Number Line Rudin, in Principles of Mathematical Analysis, defines an ordered set S as a set with a relation such that (i) If x\in S the one and only one of the statements$$x<y,\,\,\,\,x=y\,\,\,\,\,y&...
Let $(\mathbb{K},0,1,+,\times,\leq)$ be an ordered field. Let $\delta$ be its cofinality, i.e. the length of the smallest sequence of $\mathbb{K}$ that is cofinal with it. $\delta$ is a regular ...