nLab real closed field

Definition

An ordered field $F$ is real closed if it satisfies the following two properties:

• Any non-negative element $x \geq 0$ in $F$ has a square root in $F$;

• Any odd-degree polynomial function with coefficients in $F$ has a root in $F$.

Notice that the order on a real closed field is definable from the algebraic structure: $x \leq y$ if and only if $\exists_z x + z^2 = y$. (In particular, there is a unique ordering on a real closed field, defined by taking the positive elements to be precisely the nonzero squares.) In fact, the category of real closed fields and order-preserving field homomorphisms is a full subcategory of the category Field of fields and field homomorphisms.

Properties

Real closed fields can be equivalently characterized by any of the following properties:

1. $F$ is not algebraically closed, but some finite extension is. This extension is necessarily $F[\sqrt{-1}]$. See also fundamental theorem of algebra.

2. As a field, $F$ is elementarily equivalent to the field of real numbers.

3. The intermediate value theorem holds for all polynomial functions with coefficients in $F$.

4. $F$ is an ordered field that has no ordered algebraic extension.

In fact, there is a completion of any ordered field to a real closed field, in the following sense:

Theorem

The full inclusion of the category of real closed fields and field homomorphisms to the category of ordered fields and ordered field homomorphisms has a left adjoint.

Proof

We give a brief sketch of proof, referring to Lang’s Algebra ($3^{rd}$ edition), section IX.2, for more details.

First, for each ordered field $F$, there is a real closed algebraic extension $F \to R$ that is order-preserving (theorem 2.11). This is called a real closure of the ordered field $F$.

Second, any two real closures of $F$ are uniquely isomorphic (theorem 2.9); in fact, the proof shows there is at most one order-preserving homomorphism over $F$ between any two real closures. Therefore we may speak of the real closure of $F$, which we denote as $\widebar{F}$.

Finally, let $F \to R$ be any order-preserving field homomorphism to a real closed field $R$. We must show that $F \to R$ extends uniquely to a homomorphism $i: \widebar{F} \to R$. Any such homomorphism $i$ must factor through the subfield $R' \hookrightarrow R$ consisting of elements $\alpha \in R$ that are algebraic over $F$, since $\widebar{F}$ is algebraic over $F$. But this subfield is also real closed. Therefore, by the preceding paragraph, there is at most one homomorphism $\widebar{F} \to R'$ extending $F \to R'$, and the proof is complete.

Examples

1. The real numbers form a real closed field.

2. Real algebraic numbers form a real closed field, which is the real closure of the ordered field of rational numbers.

3. A field of nonstandard real numbers (as in Robinson nonstandard analysis) is real closed.

4. Surreal numbers form a (large) real closed field.

5. If $F$ is real closed, then the field of Puiseux series over $F$ is also real closed.

6. More generally, given a real closed field $F$, the field of Hahn series over $F$ with value group $G$ (a linearly ordered group) is real closed provided that $G$ is divisible.

7. Any o-minimal ordered ring structure $R$ is a real closed field.

8. Given an o-minimal ordered ring $R$, the field of germs at infinity of definable functions $R \to R$ in any o-minimal expansion of $(R, 0, 1, +, -, \cdot, \lt)$ is real closed. (By “germ at infinity”, we mean an equivalence class of functions for which $f \equiv g$ if and only if $f(x) = g(x)$ for all sufficiently large $x$.)

Infinites and infinitesimals

Each real closed field $R$ contains a valuation subring $B \hookrightarrow R$ consisting of the “bounded” or archimedean elements, i.e., elements $x \in R$ such that $-n \leq x \leq n$ for some integer multiple $n$ of the identity. An element in the complement of $B$ is an infinite element of $R$, and the reciprocal of an infinite element is an infinitesimal element. The field of fractions of $B$ is clearly $R$.

We remark that any real closed field contains a copy of the field of real algebraic numbers, which as before we denote by $\widebar{\mathbb{Q}}$ (not to be confused with the algebraic closure of $\mathbb{Q}$). Each of the elements of $\widebar{\mathbb{Q}}$ is archimedean.

Let $B^\ast$ be the group of units of $B$. The quotient $R^\ast/B^\ast$ is the value group of $R$. It can be viewed as the “group of orders of infinities and infinitesimals” of $R$. If $R$ is real closed, then the value group is a linearly ordered divisible group (divisible because we can take $n^{th}$ roots of positive elements in $R$). The structure of the value group as ordered group is an important invariant of the real closed field.

In the other direction, to each ordered divisible abelian group $G$, there exists a real closed field having $G$ as its value group. For example, one may form the Hahn series over $\widebar{\mathbb{Q}}$ with value group $G$.

In constructive mathematics

In constructive mathematics, in order to define an odd degree polynomial, one has to use the tight apartness relation $x \# y$ or $x \neq y$ of an ordered field, defined as $x \lt y \vee x \gt y$, instead of denial inequality $\neg (x = y)$, which is not normally related to the order of the ordered field. A polynomial $p = \sum_{i \leq n} a_n z^n$ has an odd degree if $n$ is odd and $a_n$ is apart from zero.

In addition, the original definition of an ordered field bifurcates into two. One has to decide whether to use mere existence of a root in the sense of traditional first-order logic or constructive existence in the sense of the BHK interpretation in the formulation of a real closed fields:

Proposition

An ordered field $F$ is a real closed field if for every non-negative element $x \geq 0$ in $F$ there exists a square root in $F$, and for any odd-degree polynomial function with coefficients in $F$ there exists a root in $F$.

Proposition

An ordered field $F$ is a real closed field if for every non-negative element $x \geq 0$ in $F$ there exists a square root in $F$, and for any odd-degree polynomial function with coefficients in $F$ there exists a root in $F$.

Note here the theorems do not consider the notion of constructing a specified square root in the BHK interpretation for non-negative elements of $F$, or the stronger notion of the squaring function $x \mapsto x^2$ having a principal square root on the non-negative elements of $F$. This is because the two notions are equivalent to each other for ordered fields $F$:

Theorem

The conditions that “for every non-negative element $x \geq 0$ in $F$ there exists a square root in $F$” and “$F$ has a principal square root partial function defined on the non-negative elements of $F$” are equivalent to each other.

Proof

The squaring function $x \mapsto x^2$ is an injection on non-negative elements of $F$ because it is monotonic on non-negative elements of $F$, and the first condition says that the squaring function $x \mapsto x^2$ is surjective on non-negative elements of $F$ and the second condition says that the squaring function $x \mapsto x^2$ has a section on non-negative elements of $F$; these are equivalent conditions for injective functions.

In addition, the various equivalent definitions and properties of an ordered field are no longer provably equivalent in constructive mathematics:

1. For every non-negative element $x \geq 0$ in $F$ there exists a square root in $F$, and for any odd-degree polynomial function with coefficients in $F$ there exists a root in $F$.

2. For every non-negative element $x \geq 0$ in $F$ there exists a square root in $F$, and for any odd-degree polynomial function with coefficients in $F$ one can construct a specified root in $F$.

3. For every non-negative element $x \geq 0$ in $F$ there exists a square root in $F$, and any odd-degree polynomial function $p(z)$ with degree $n$ coefficients in $F$ can be factored into a linear function $c_1 z + c_0$ with $c_0$ and $c_1$ in $F$ and a monic polynomial function $q(z)$ of degree $n - 1$ with coefficients in $F$, $p(z) = (c_1 z + c_0) q(z)$.

4. The intermediate value theorem holds for all polynomial functions with coefficients in $F$.

5. $F$ is an ordered field that has no ordered algebraic extension.

6. $F$ is not algebraically closed, but the finite extension $F[\sqrt{-1}]$ is.

7. As a field, $F$ is elementarily equivalent to the field of real numbers.

As a result, the any one of these can be used in the definition of a real closed field.

Real closure of the real numbers

For any field of real numbers, it is provable that the real numbers are a real closed field in the sense that for all non-negative real numbers there exist a real square root, and for every odd degree polynomial functions on the real numbers, there exist a root of the polynomial function. However, the real numbers themselves are not provably a real closed field in the sense that for any odd-degree polynomial function with coefficients in the reals one can construct a specified root in the reals. To show that this is the case, we turn to reframing the definitions of a real closed field in terms of surjectivity and having right inverses of real polynomial functions so that one can show that the equivalence of the definitions is equivalent to a weak form of choice.

Let $p(z) = \sum_{i \leq n} a_n z^n$ be a polynomial function on the real numbers. Every such $p(z)$ can be written as the sum of a constant $a_0$ and a polynomial function $q(z) = \sum_{1 \leq i \leq n} a_n z^n$ with a fixed point at zero. As a result, the statement that there exists a real number $z$ such that $p(z) = 0$ is equivalently the statement that there exists a real number $z$ such that $q(z) = b$, where $b = -a_0$. Thus, one can rewrite one of the conditions in the first definition of a real closed field as:

Definition

Given a real number $b$ and an odd degree polynomial function $q(z)$ on the real numbers such that $q(0) = 0$, there exists a real number $c$ such that $q(c) = b$.

Or equivalently

Theorem

Every odd degree polynomial function $q(z)$ on the complex numbers such that $q(0) = 0$ is surjective.

One can do the same analysis with the BHK interpretation of the definition of a real closed field, the statement that one can construct a specified real number $z$ such that $p(z) = 0$ is equivalently the statement that one can construct a specified real number $z$ such that $q(z) = b$, where $b = -a_0$. Thus, one can rewrite the fundamental theorem of algebra as follows:

Theorem

Given a real number $b$ and a odd degree polynomial function $q(z)$ on the real numbers such that $q(0) = 0$, one can construct a specified real number $c$ such that $q(c) = b$.

Or equivalently

Theorem

Every odd degree polynomial function $q(z)$ on the real numbers such that $q(0) = 0$ has a section.

Thus, the gap between proving that the real numbers are a real closed field using mere existence and that the real numbers are real closed field using constructive existence is precisely this weak version of the axiom of choice:

Proposition

Every surjective polynomial function on the real numbers with odd degree and a fixed point at zero has a section.

In particular, consider the map $z \mapsto z^3 - 3z$. This map is surjective on the real numbers, but cannot be proven to have a section. In fact, in certain topoi, such as sheaves over $\mathbb{R}$, one can prove that there are no such sections of $z \mapsto z^3 - 3z$ on the real numbers, since any since any section on $z \mapsto z^3 - 3z$, if it exists, has to be discontinuous somewhere in the closed interval $[-2, 2]$, and in those topoi all functions on the real numbers are continuous.

The unprovability of this weak version of choice in neutral constructive mathematics also implies that one cannot factor every odd degree real polynomial function $p(z)$ of degree $n$ into an affine function $c_1 z + c_0$ with real numbers $c_0$ and $c_1$ and a monic real polynomial function $q(z)$ of degree $n - 1$, $p(z) = (c_1 z + c_0) q(z)$, since one would need to first construct the specified real root $-\frac{c_0}{c_1}$ of $p(z)$. Being able to factor polynomial functions is required to show that the complex numbers are an algebraically closed field.

Related concepts

• algebraic closure

• fundamental theorem of algebra

• Euclidean geometry – Tarski’s axioms

References

• Serge Lang: Algebra, 3rd ed. Springer (2002) [doi:10.1007/978-1-4613-0041-0]

• David Marker, Notes on Real Algebra (link)