An ordered field $F$ is real closed if it satisfies the following two properties:
Any positive 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 polynomials 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 $G$.
Serge Lang, Algebra (3rd edition), Addison-Wesley, 1993.
David Marker, Notes on Real Algebra (link)