real closed field

Assume from now on that is a real closed field.

More generally, any real closed field has Brauer group of order two.

However, there are no primitive 3rd roots of unity in a real closed field.

There are many different characterizations of real closed fields.

The computable real numbers form a real closed field.

A real closed field is a field F in which any of the following equivalent conditions are true:

We have therefore the following invariants defining the nature of a real closed field F:

Real closed fields are unstable, as they are infinite and have a definable total order.

Axioms admitting of a (multi-dimensional) faithful interpretation as a real closed field.

RCF, the theory of real closed fields.

The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.

The theory of real closed fields was invented by algebraists, but taken up with enthusiasm by logicians.

Similarly in the class of ordered fields, the existentially closed structures are the real closed fields.

The most important model theoretic consequences hereof: The theory of real closed fields is complete, o-minimal and decidable.

Conversely, every subring of a product of real closed fields with this property is real closed.

Quantifier elimination on real closed fields takes a doubly-exponential time (see Cylindrical algebraic decomposition).

In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers.

The ordered field of real algebraic numbers is the unique atomic model of the theory of real closed fields.

This is equivalent to the decidability of real closed fields, of which elementary Euclidean geometry is a model.

A formally real field with no formally real proper algebraic extension is a real closed field.

If the continuum hypothesis holds, all real closed fields with cardinality the continuum and having the η property are order isomorphic.

In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field.

Nash functions and manifolds can be defined over any real closed field instead of the field of real numbers, and the above statements still hold.

Some first-order theories are algorithmically decidable; examples of this include Presburger arithmetic, real closed fields and static type systems of many programming languages.

For every prime ideal p of A, the residue class ring A/p is integrally closed and its field of fractions is a real closed field.