Really closed body

In algebra , the real closed bodies are bodies that have some essential properties in common with the field of real numbers : For example, polynomials with odd degrees always have a zero there and these bodies can be given an order relation that is clearly determined by the body structure which they become into ordered bodies .

A real closed body is at most among the formally real bodies , those are the bodies on which a structurally compatible order can be defined at all: Every real algebraic body expansion destroys the possibility of arranging the real closed body. At the same time it is "almost" algebraically closed: Every real algebraic field expansion makes it an algebraically closed field .

The mathematical concept described here, which in addition to the concept of the real closed body, has also given rise to concepts such as formally real bodies , Pythagorean bodies and Euclidean bodies , describes certain properties of real numbers algebraically and uses such descriptions for the axiomatic definition of a class of bodies with these properties .

definition

A field is called real closed if one of the following equivalent conditions applies: It is formally real and ${\displaystyle K}$

1. none of its real algebraic extensions are formally real,
2. the body expansion is algebraically closed, ${\displaystyle K(i)}$
3. every two-dimensional field extension is algebraically closed,
4. every real finite-dimensional field extension is algebraically closed.

Applications of the concept

When defining the real closed fields, two essential properties of the real numbers are taken into account:

• Arrangement:

1. The real numbers allow an arrangement with which they become an ordered body.
2. There is only one arrangement with this property.

• Maximumity or isolation:

1. If one extends the real numbers , the possibility of arrangement is lost. ${\displaystyle \mathbb {R} (i)=\mathbb {C} }$
2. All real algebraic extensions lead to algebraic closure.

Bodies that allow an arrangement, i.e. share the first arrangement property with the real numbers, are formally real , a purely algebraic definition is:

A field is called formally real if −1 cannot be represented as a finite sum of squares, i.e. h .: There are no elements with . → For a more detailed description of these bodies, see Ordered Body . ${\displaystyle K}$ ${\displaystyle y_{1},\ldots ,y_{n}\in K}$ ${\displaystyle -1=y_{1}^{2}+\ldots +y_{n}^{2}}$

For every body that allows exactly one arrangement, this can be described purely algebraically by the following definition:

${\displaystyle a<b}$applies if and only if is a square number, i.e. has a solution in the body. ${\displaystyle b-a}$ ${\displaystyle b-a=X^{2}}$ ${\displaystyle x\neq 0}$

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

The real numbers have the property that the special field extension makes any arrangement as an ordered field impossible. They share this property with every formally real body, since a body can never be arranged if the square classes of −1 and 1 coincide in it. It is interesting here which algebraic expansions can still be carried out without −1 becoming a square number and thus no more arrangement is possible: ${\displaystyle \mathbb {R} (i)}$

• Since a real closed field is a maximal field with the property that it allows an arrangement, every algebraic extension destroys this property.
• A Euclidean field is only required (apart from the fact that it must have the characteristic 0) that it contains exactly the two square classes of −1 and 1. Here the possibility of arrangement is not destroyed by every, but every two-dimensional body extension.

properties

• A formally real field is never algebraically closed, because in algebraically closed fields −1 is a square,
• every formally real body has the characteristic 0 and contains an infinite number of elements,
• Lower bodies of formally real bodies are formally real again.
• A real closed body can be made into an ordered body by exactly one order relation. The positive elements are exactly the squares. All finite sums of squares are also squares and therefore positive. Therefore every real closed body is Pythagorean.
• A real closed body is always a Euclidean body, a Euclidean body always a formally real Pythagorean body.
• The only body automorphism of a real closed body is the identical mapping .
• If, in the case of a finite-dimensional body expansion, the body is real closed, then the expansion is Galois if and only if is. ${\displaystyle K<L}$ ${\displaystyle L}$ ${\displaystyle K=L}$

Examples and counterexamples

• The field of complex numbers is not formally real and therefore not a real closed field. ${\displaystyle \mathbb {C} }$
• The field of real numbers is real closed. Obviously −1 is not the sum of squares and is the only real algebraic expansion field that is not formally real according to the previous example. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$
• The field of the rational numbers is formally real, but not real closed, because the field is a real, algebraic, formally real extension. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} ({\sqrt {2}})=\{a+b{\sqrt {2}};\,a,b\in \mathbb {Q} \}}$
• The field of real and over algebraic numbers is real closed. ${\displaystyle \mathbb {Q} }$
• Every real closed body is Euclidean , every Euclidean formally real.

Existential sentences

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

• If formally real and an algebraically closed upper body, there is an intermediate field with , where a root of −1 is. is then a real closed body extension of . ${\displaystyle K}$ ${\displaystyle {\overline {K}}}$ ${\displaystyle K\subset L\subset {\overline {K}}}$ ${\displaystyle {\overline {K}}=L(i)}$ ${\displaystyle i}$ ${\displaystyle L}$ ${\displaystyle K}$

Applying this theorem to the smallest algebraic closure we get:

• Every formally real field has an algebraic and real closed extension.

This statement can be tightened considerably for arranged bodies:

• Be an ordered body. Then, apart from isomorphism, there is exactly one algebraic and real-closed continuation, the unambiguous arrangement of which continues the order of . ${\displaystyle K}$ ${\displaystyle K}$

For the construction one adjoins all square roots of positive elements of and shows that the resulting body is formally real. Then you apply the above theorem and get an algebraic and real closed extension, from which you then have to show the uniqueness statement. In the case of an arranged body, one can speak of the real closure . ${\displaystyle K}$

literature

• Thomas W. Hungerford: Algebra. 5. print. Springer-Verlag, 1989, ISBN 0-387-90518-9
• Serge Lang : Algebra. Reading, Mass. Addison-Wesley, 1965
• Saunders MacLane and Garrett Birkhoff : Algebra , New York: The Macmillan Company, 1967
• Van der Waerden : Algebra I , Springer Verlag, ISBN 3-540-56799-2