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 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 of fields and field homomorphisms.
Real closed fields can be equivalently characterized by any of the following properties:
$F$ is not algebraically closed, but some finite extension is. This extension is necessarily $F[\sqrt{-1}]$. See also fundamental theorem of algebra.
As a field, $F$ is elementarily equivalent to the field of real numbers.
The intermediate value theorem holds for all polynomial functions with coefficients in $F$.
$F$ 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:
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.
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.
The real numbers form a real closed field.
Real algebraic numbers form a real closed field, which is the real closure of the ordered field of rational numbers.
A field of nonstandard real numbers (as in Robinson nonstandard analysis) is real closed.
Surreal numbers form a (large) real closed field.
If $F$ is real closed, then the field of Puiseux series over $F$ is also real closed.
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.
Any o-minimal ordered ring structure $R$ is a real closed field.
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 $x$.)
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, the various definitions of real closed field bifurcate, because of different definitions of an odd-degree polynomial. One could define an odd-degree polynomial as a polynomial whose coefficient for an odd number $n$ is not equal to zero and whose coefficient for all $i \gt n$ for is equal to zero. On the other hand, one could also define an odd-degree polynomial as a polynomial whose coefficient for an odd number $n$ is apart from zero and whose coefficient for all $i \gt n$ for is equal to zero. These two definitions are different from each other, the real numbers satisfy the second, while they do not satisfy the first.
Serge Lang, Algebra (3rd edition), Addison-Wesley, 1993.
David Marker, Notes on Real Algebra (link)
Last revised on June 2, 2022 at 18:18:36. See the history of this page for a list of all contributions to it.