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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.

Let $F$ be a field, then $F$ is an ordered field if there is a binary relation $\leq$ on $F$ with the following properties:

  1. $\leq$ is a linear (total) order $F$.
  2. If $x\leq y$ then $x+z\leq y+z$.
  3. If $0\leq x,y$ then $0\leq ab$.

For example, $\Bbb Q$ and $\Bbb R$ are ordered fields.

Not every field can be ordered. For example, one can verify that if $F$ is an ordered field then the characteristics of $F$ is zero.

If $F$ is an ordered field then all the squares are positive, and so are their sums. Since $-1$ is not positive we have that in an ordered field, $-1$ is not the sum of squares, and therefore algebraically closed fields (such as $\Bbb C$) are not ordered fields.

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Is the multiplicative structure of a totally ordered field unique?

Is it possible to find totally ordered fields $K$ and $L$, and a map $f: K \to L$ that is an isomorphism of the ordered additive group structures such that $f(1)=1$, but which is not an isomorphism of fields? I know that this is impossible if $K$ or…
user120974
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Completeness of $\Bbb R(x)$?

I recently claimed somewhere that any two complete ordered fields are isomorphic. Someone pointed out that here we need to distinguish between Cuachy completeness (every Cauchy sequence converges) and Dedekind completeness (any set bounded above has…
David C. Ullrich
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Is Nested Interval Property $\iff$ Order Completeness in any Ordered Field?

Let $R$ be an arbitrary Ordered Field. Order Completeness Property: If $S \subset R$ is bounded above, then $\exists \ c \in R$ that is an upperbound of S and for every upperbound $b$ of S, we have $c \leq b.$ Nested Interval Property: If…
Saikat
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Product of infinitesimal by inverse of non infinitesimal is infinitesimal

This is probably very simple, though I am getting confused because of my inexperience with infinitesimals. Let $K$ be an ordered real-closed field. We say that $a \in K$ is finite if it's absolute value is less than $n$ for some positive integer…
user480840
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$R((x))$ Set of formal Laurent Series is an ordered field which is Cauchy Complete.

Let $R$ be any ordered field. Define $R((x)):=\{\sum\limits_{k=-n}^\infty a_kx^k:a_k \in R \ , \ n \in\mathbb N\}$ Addition: $\sum_\limits{k=-n}a_kx^k+\sum_\limits{k=-m}b_kx^k:= \sum_\limits{k=-max\{n,m\}}…
Saikat
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Every cofinal real closed subfield is dense?

Consider a real closed subfield $F_0$ of a real closed field $F_1$ that is cofinal. Here, "$F_0$ is cofinal in $F_1$" means "For each $x \in F_1$, there exists $y \in F_1$ such that $x < y$". Then, is $F_0$ order dense in $F_1$?
Yushiro Aoki
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$K$ is an ordered field $\leftrightarrow$ Subset $P \subseteq K$ exists with certain characteristics

I need help for the following task. Unfortunately I don't have any idea how to start. a) Prove: A field $K$ is an ordered $\leftrightarrow$ Subset $P \subseteq K$ exists with following features: $\forall x,y \in P$ : $x+y \in P$ $\forall x,y \in P…
Analysis_Mark
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Topological Connectedness $\implies$ Order Completeness

Let $\mathbb{F}$ be an arbitrary ordered field. We define: $\textbf{Topological Connectedness:}$ If $\mathbb{F}=A \cup B$ where A,B are nonempty and open, then $A \cap B \neq \phi$. $\textbf{Order Completeness :}$ If $S \subset \mathbb{F}$ is…
Saikat
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A corollary of the Nested Interval Property.

Let $R$ be an ordered field with the Nested Interval Property: If $I_1,I_2,...,I_n,...$ be a collection of nested closed intervals, i.e $I_1 \supseteq I_2 \supseteq ... \supseteq I_n \supseteq ...$ then $\bigcap_{1}^{\infty} I_i \neq \phi$ where…
Saikat
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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 definition - it's quite clean - but I'm not convinced it is…
joeb
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Prove the following set has a multiplicative inverse

Let $F = \{x + y\sqrt{7} : x, y \in Q\}$ with the usual addition and multiplication operations. Show that $F$ has a multiplicative inverse. So far, I have $$\left(x_1 + y_1\sqrt{7}\right) \left(\frac{1}{x_1} +…
Don Larynx
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Less then or Equal to Relation on Ordered Fields

If we consider the natural numbers, we have for all $a,b\in\mathbb{N}$ that $a\leq b\Longleftrightarrow (\exists x\in\mathbb{N})[a+x=b]$. (For example if $a=2$ and $b=5$ there exists $3\in\mathbb{N}$ such that $2+3=5$.) Now, as a generalization of…
PAT
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Show that a field of rational expressions with real coefficients is ordered.

I need help for this task, that I just can't solve since I have no idea where to start. It is connected with this task. Let $K$ be the field of rational expressions with real coefficients i.e. expressions of the form $\frac{p}{q}$ where $p$ and…
Analysis_Mark
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Subfield of $\mathbb{R}$ that satisfies the Ratio Test must contain every reals.

I am reading James Propp's Real Analysis in reverse. There in pg. 12 he shows Ratio Test $\implies$ LUB Axiom. First he shows that Ratio Test $\implies$ Archimedean Property: By Consider the following sequence $\left(\frac{1}{2^n}\right)$…
Saikat
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Is Nested Interval Property $\iff$ Monotone Convergence Property in an ordered field?

Given any ordered field $R$ Can we conclude Monotone convergence Property $\iff$ Nested Interval Property? Monotone Convergence Property: Any monotone increasing sequence bounded above in $R$ is convergent in $R.$ Nested Interval Property: If…
Saikat
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