I have a result that uses ordered fields. However, I am ignorant of the literature surrounding ordered fields. I have read the basic facts about them, such as those contained in the wikipedia page. However, in the introduction, I would like to provide some motivation.

My result has the property that if it holds for some ordered field $k$, then it holds for any ordered subfield of $k$. Does that mean it is enough to prove my result for real-closed ordered fields?

I also read the statement that to prove a 1st order logic statement for a real-closed ordered field, then it is enough to prove it for one real-closed ordered field, such as $\mathbb{R}$ for instance. Can someone please provide a reference for that?

A mathematician mentioned to me a classical link between iterated quadratic extensions and ordered fields. What is the precise statement please?

I would also like a number of interesting examples of ordered fields. I know for example of an interesting non-archimedean example using rational functions (that I have learned about from the wikipedia page on ordered fields).

Does anyone know of a survey on ordered fields, or a reference containing answers to my questions above?


A few remarks:

I don't know of a systematic study of ordered fields with many examples, but I can provide a few references as well as a list of types of ordered fields. For a general algebraic approach to ordered fields, you can look up the chapter "Ordered fields" in Serge Lang's Algebra. For notes about formally real fields or orderable fields, you can look at the work of Artin and Schreier. Most introductive books to model theory and its applications to algebra have a chapter about real closed fields.

Also, the book Super-real fields of Dale and Woodin studies set-theoretic properties of a class of real closed fields vaguely related to non-standard analysis, but this is not really something you need to look into unless you're interested in set theory.

As for gaining perspective on ordered fields, for good presentations of the role of different types of ordered fields in the conception of infinitesimals and continua, I suggest you look into the work of Philip Ehrlich, e.g. this one and here.

Examples of ordered fields

Here is a list of important types of ordered fields that are most frequently encountered.

-Archimedean ordered fields, in particular $\mathbb{R}$, $\mathbb{Q}$, and the field $\mathbb{A}$ of real algebraic numbers.

-Given a linear order $E$, and an ordered field $F$, the field $F(E)$ of fractions of polynomials with indeterminates in $E$, see here in the case when $F=\mathbb{Q}$. In particular, the field $F(X)\cong F(1)$ of rationnal functions with coefficients in $F$.

-Given a linearly ordered abelian group $G$, and an ordered field $F$, the field $F[[G]]$ of Hahn series with value group $G$ and residue field $F$.

-Given an ordered field $F$ and a free ultrafilter $\mathcal{U}$ on an infinite set $I$, the ultrapower $^*F_{\mathcal{U}}$ of $F$ modulo $\mathcal{U}$, and especially the important case when $F=\mathbb{R}$ and $I=\mathbb{N}$.

-Given a $\varepsilon$-number $\lambda$, the field $\mathbf{No}(\lambda)$ of surreal numbers with birthdate strictly below $\lambda$, see here.

-Fields of real valued functions, in particular Hardy fields, see here for a perspective on those with respect to real asymptotic differential algebra.

-Fields of transseries, in particular logarithmic-exponential transseries, see here (careful: this article is about a different type of transseries than that mentionned in the Wikipedia article) or here (first preprint in the list).

Two functorial constructions

Important constructions within the class of ordered fields are the real closure and the Cauchy completion. Most of the examples above are already real-closed but few are Cauchy-complete, so this produces new examples of ordered fields.

The real closure construction $F_{real}$ and Cauchy completion construction $F_{Cauchy}$ are functors, in contrast with the algebraic closure for fields of a given characteristic). The corresponding categories are reflective within the category of ordered fields with certain types of morphisms. Moreover, we have $F_{Cauchy} \circ F_{real} \cong F_{real} \circ F_{Cauchy} \circ F_{real}$ (i.e. the Cauchy completion of a real-closed field remains real-closed). This makes it is easy to manipulate them in conjunction.

Universal formulas

In model theory, given a first order language, a universal formula is a formula $\phi[\overline{x}]$ of the form $\forall \overline{u}(\theta[\overline{x},\overline{u}])$ where $\theta[\overline{x},\overline{u}]$ is quantifier-free.

Given a theory $T$, a formula $\varphi[u]$ is said logically equivalent to a universal formula modulo $T$ if there is a quantifer-free formula $\phi[\overline{x}]$ such that $T \vdash \forall \overline{x}(\varphi[\overline{x}] \leftrightarrow \phi[\overline{x}])$. It is equivalent that $\varphi[\overline{x}]$ be preserved by substructures, i.e. that for all models $\mathcal{M}\subseteq \mathcal{N}$ of $T$ and $\overline{a} \in M^n$, that $\mathcal{M} \vDash \varphi[\overline{a}]$ be equivalent to $\mathcal{N} \vDash \varphi[\overline{a}]$.

In particular, if $\varphi$ is a such a sentence in the language of ordered fields which one can prove for real closed fields, then since every ordered field embeds in a real-closed field (in particular, in its real closure), it is true in any ordered field.

By Tarski's work, the theory of real closed fields is elementary and complete, so it suffices to prove a first order result in $\mathbb{R}$ (or any other real closed field, but $\mathbb{R}$ is quite unique) to derive it in any ordered field. You will find Tarksi's result in any model theory book mentioning real closed fields.

Notice that if you take a sufficiently big cardinal $\kappa$ and a free ultrafilter $\mathcal{U}$ on $\kappa$, then $^*\mathbb{R}_{\mathcal{U}}$ contains any ordered field you want, so you can prove this only in this case. Likewise, $\mathbf{No}(\kappa)$ contains all ordered fields if $\kappa$ is sufficiently big.

Quadratic extensions

If a formally real field $k$ is such that for $a \in k^{\times}$, either $a$ is a square or $-a$ is a square (but not both), then it is uniquely ordered by saying that an element is positive if it is a square. In fact this is an equivalence, since given $a$ in an ordered field $F$ such that neither $a$ nor $-a$ are squares, the field $F[\sqrt{a}]$ admits two orders: one where $\sqrt{a}>0$ and one where $\sqrt{a}<0$. If one iterates quadratic extensions by square roots of elements $a$ such that $-a$ has no square root, then one obtains such a field with exactly one compatible (and easily defined) order. This may not be the classical result you were told about.

  • $\begingroup$ I am still going through the examples. Thank you. Regarding your paragraph on quadratic extensions, do you assume that $k$ is of characteristic $0$? The reason I ask is because $\mathbb{F}_3$ or $\mathbb{F}_7$ for example satisfy the property that for $a \in k^\times$, either $a$ is a square or $-a$ is a square. One can define an order as you do, but it would not be an "ordered field", right? $\endgroup$ – Malkoun Dec 17 '18 at 10:14
  • $\begingroup$ Finally, in the section on quadratic extensions, do you mean perhaps "...one where $\sqrt{a} > 0$ and one where $\sqrt{a} < 0$? $\endgroup$ – Malkoun Dec 17 '18 at 10:16
  • $\begingroup$ You're right in both cases, I will edit that soon. (in the first mistake: the positive cone won't be stable under sum in non-zero characteristic, in fact the field must be formally real) $\endgroup$ – nombre Dec 17 '18 at 10:53

This answer complements the one of @nombre. For a quick review of non-Archimedean ordered fields as a particular case of non-Archimedean valued fields see Summary on non-Archimedean valued fields where you can find a variety of ordered fields and references for several characterizations.

For references that prove your statement regarding 1st order logic, see Theorem 3.22. In particular, the two last references of that theorem.

  • $\begingroup$ It is nice to see that there is such a review for valued fields. Are you one of the authors? $\endgroup$ – nombre Dec 16 '18 at 23:00
  • $\begingroup$ I am the first author. Indeed, I searched for a review like that in the literature and I did not find one so with K. Shamseddine we made our own review. When it was completed it was nice to have it. $\endgroup$ – Chilote Dec 17 '18 at 2:16
  • $\begingroup$ I apologize that I cannot accept both answers, but I do thank you both, you and @nombre. $\endgroup$ – Malkoun Dec 17 '18 at 9:21

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