Real closed field

The real closed fields are in algebra bodies that have in common with the field of real numbers some essential properties: For example, polynomials have odd degree there is always a zero and these bodies can be equipped with a uniquely determined by the body structure order relation, with which they are to parent bodies.

A real closed field is maximal among the formally real bodies, which are the body, where ever a structure compatible order can be defined: Every real algebraic field extension destroys the possibility to arrange the real closed fields. At the same time he is " almost" algebraically closed: Every real algebraic field extension makes it an algebraically closed field.

The mathematical approach described here, which has spawned terms such as formal real body, Pythagorean body and Euclidean body in addition to the concept of a real closed field, describes certain properties of real numbers algebraically and use such descriptions to the axiomatic definition of a class of objects with these properties.

Definition

A body is called real closed if one of the following equivalent conditions is true: it is formally real and

Applications of the concept

In the definition of real closed fields two essential properties of real numbers are taken into account:

• Arrangement:

• Maximality or seclusion:

Body, which allow a array, so share the first array property with the real numbers are called, formally real, a purely algebraic definition is:

For each body, which allows exactly one arrangement, it can be described purely algebraically by the following definition:

In other words: A number is exactly positive if it lies in the square class of 1. The existence of exactly one arrangement is equivalent to the fact that exactly two square classes, the 1 and -1 are contained in the body of characteristic 0. → A body that can be arranged to exactly one type is called a Euclidean body.

The real numbers have the property that the special body extension makes any arrangement impossible as ordered field. They share this property with any formally real field, since a body can never be ordered if collapse the square classes of -1 and 1 in him. It is interesting that algebraic extensions at all are feasible without -1 for square number and thus no arrangement is possible:

• Since a real closed field is a maximum body with the property that it permits an arrangement destroys any algebraic extension this property.
• From is a Euclidean body ( apart from the fact that he must have the characteristic 0) only required that he has exactly two square classes of -1 and 1. Here, the arrangement possibility is not destroyed by any, but by each two-dimensional field extension.

Properties

• A formally real body is never algebraically closed, because in algebraically closed fields is -1 a square,
• Any formally real body has the characteristic 0 and contains infinitely many elements
• Lower body formally real bodies are formally real again.
• A real closed field can be made by exactly one order relation to a parent body. The positive elements are exactly the squares. Also, all finite sums of squares are squares and so positive again. Therefore, any real closed field is Pythagorean.
• A real closed field is always a Euclidean body, a Euclidean body always a formally real Pythagorean body.
• The only Körperautomorphismus a real closed field is the identical figure.
• If in a finite field extension of the body completed real, then the extension is Galois if and only if is.

Examples and counter-examples

• The field of complex numbers is not formally real and so no real closed field.
• The field of real numbers is complete real. Apparently, not -1 sum of squares and the only real algebraic extension field is, which is not formally real by the previous example.
• The field of rational numbers is formally real, but not real closed, because the body is a real algebraic, formally real extension.
• The field of real algebraic numbers is complete and real.
• Every real closed field is Euclidean, Euclidean every formally real.

Existence theorems

First, one can show with the help of the existence of the algebraic statements that every formally real body has a real closed upper body:

• Is formally real and an algebraically closed upper body, then there's an intermediate body, wherein a root of -1. is then a real closed field extension of.

By applying this rate to the smallest algebraic degree, one obtains:

• Every formally real algebraic and real body has a closed extension.

For ordered fields can significantly exacerbate this statement:

• Be an ordered field. Then there are up to isomorphism exactly one algebraic and real - closed extension whose unique arrangement, the order of continues.

To construct one adjoint square roots of all positive elements of and shows that the so- formed body is formally real. Then you turn the sentence above and receives a real algebraic and completed expansion, from which one has then to show the uniqueness statement. In case of an ordered field so you can talk about the real degree.