An ordered field is real closed if it satisfies the following two properties:
Any positive element in has a square root in ;
Any odd-degree polynomial with coefficients in has a root in .
Notice that the order on a real closed field is definable from the algebraic structure: if and only if . (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:
As a field, is elementarily equivalent to the field of real numbers.
The intermediate value theorem holds for all polynomials with coefficients in .
is an ordered field that has no ordered algebraic extension.
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 ( edition), section IX.2, for more details.
First, for each ordered field , there is a real closed algebraic extension that is order-preserving (theorem 2.11). This is called a real closure of the ordered field .
Second, any two real closures of are uniquely isomorphic (theorem 2.9); in fact, the proof shows there is at most one order-preserving homomorphism over between any two real closures. Therefore we may speak of the real closure of , which we denote as .
Finally, let be any order-preserving field homomorphism to a real closed field . We must show that extends uniquely to a homomorphism . Any such homomorphism must factor through the subfield consisting of elements that are algebraic over , since is algebraic over . But this subfield is also real closed. Therefore, by the preceding paragraph, there is at most one homomorphism extending , and the proof is complete.
The real numbers form a real closed field.
A field of nonstandard real numbers (as in Robinson nonstandard analysis) is real closed.
Surreal numbers form a (large) real closed field.
If is real closed, then the field of Puiseux series over is also real closed.
Any o-minimal ordered ring structure is a real closed field.
Given an o-minimal ordered ring , the field of germs at infinity of definable functions in any o-minimal expansion of is real closed. (By “germ at infinity”, we mean an equivalence class of functions for which if and only if for all sufficiently large .)
Each real closed field contains a valuation subring consisting of the “bounded” or archimedean elements, i.e., elements such that for some integer multiple of the identity. An element in the complement of is an infinite element of , and the reciprocal of an infinite element is an infinitesimal element. The field of fractions of is clearly .
We remark that any real closed field contains a copy of the field of real algebraic numbers, which as before we denote by (not to be confused with the algebraic closure of ). Each of the elements of is archimedean.
Let be the group of units of . The quotient is the value group of . It can be viewed as the “group of orders of infinities and infinitesimals” of . If is real closed, then the value group is a linearly ordered divisible group (divisible because we can take roots of positive elements in ). 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 , there exists a real closed field having as its value group. For example, one may form the Hahn series over with value group .
Serge Lang, Algebra (3rd edition), Addison-Wesley, 1993.
David Marker, Notes on Real Algebra (link)