nLab
real closed field

Contents

Definition

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

  • Any positive element x0x \geq 0 in FF has a square root in FF;

  • Any odd-degree polynomial with coefficients in FF has a root in FF.

Notice that the order on a real closed field is definable from the algebraic structure: xyx \leq y if and only if zx+z 2=y\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.

Properties

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

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

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

  3. The intermediate value theorem holds for all polynomials with coefficients in FF.

  4. FF 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 rd3^{rd} edition), section IX.2, for more details.

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

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

Finally, let FRF \to R be any order-preserving field homomorphism to a real closed field RR. We must show that FRF \to R extends uniquely to a homomorphism i:F¯Ri: \widebar{F} \to R. Any such homomorphism ii must factor through the subfield RRR' \hookrightarrow R consisting of elements αR\alpha \in R that are algebraic over FF, since F¯\widebar{F} is algebraic over FF. But this subfield is also real closed. Therefore, by the preceding paragraph, there is at most one homomorphism F¯R\widebar{F} \to R' extending FRF \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 FF is real closed, then the field of Puiseux series over FF is also real closed.

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

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

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

Infinites and infinitesimals

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

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 *B^\ast be the group of units of BB. The quotient R */B *R^\ast/B^\ast is the value group of RR. It can be viewed as the “group of orders of infinities and infinitesimals” of RR. If RR is real closed, then the value group is a linearly ordered divisible group (divisible because we can take n thn^{th} roots of positive elements in RR). 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 GG, there exists a real closed field having GG as its value group. For example, one may form the Hahn series over ¯\widebar{\mathbb{Q}} with value group GG.

References

  • Serge Lang, Algebra (3rd edition), Addison-Wesley, 1993.

  • David Marker, Notes on Real Algebra (link)

Revised on August 21, 2011 23:24:47 by Todd Trimble (69.118.58.208)