Euclidean field

A Euclidean body is a body (in terms of the ring ), which is a parent body, and in which each non-negative element is a square root.

Every real closed field is Euclidean and Euclidean every body is a Pythagorean and formally real body.

Euclidean body play in synthetic geometry an important role: the body coordinates of a Euclidean plane is always a Euclidean body on these bodies can always build a Euclidean plane. The term " Euclidean plane " is somewhat more general than in the usual geometry, where the Euclidean plane is defined approximately by Hilbert's system of axioms of Euclidean geometry so that it is necessarily an affine plane on the special Euclidean field of real numbers - one to Hilbert system equivalent formulation in the language of linear algebra is: a Euclidean plane is an affine space, the vector space of a two-dimensional Euclidean vector space shifts, ie is isomorphic with a scalar product. The Euclidean levels of synthetic geometry are closely related to classical questions of constructability. For these problems to be solved additional axioms, such as the protractor axiom which requires the existence of a radian, and the angular distribution axiom, which can not be met in all Euclidean planes will end.

If, in the analytically formulated two - or three-dimensional, real Euclidean geometry, the real numbers as coordinates range by any Archimedean Euclidean body, then you can within these geometric models for non-Euclidean geometries construct, instead of the axioms of continuity ( Axiom Group V in Hilbert's axiom system ) satisfy the weaker axioms of the circle. In these models, the absolute geometry are then still all constructions with ruler and compass executable.

A certain importance Euclidean body as counter-examples in the theory of field extensions and Galois theory of, next in transcendence investigations in number theory.

Euclidean body and levels carry their name in honor of the ancient mathematician Euclid of Alexandria, with both their name to its axiomatic structure of - thanks Euclidean geometry in his work " The Elements" - until today named after him. → The term " Euclidean ring " from the Teilbarkeitstheorie in commutative rings is in no narrower content related to those described in this article as the terms that he is also named after Euclid, namely after the Euclidean algorithm described by him.

Alternative definitions

A Pythagorean body, ie a body in which every sum of squares is a square again is precisely then a Euclidean body if it contains exactly two square classes. Although no arrangement is by this purely algebraic definition given, but there is in such a Pythagorean bodies only only one chance to make it to a parent body and through the definition

So can these " canonical order" than by the algebraic structure view with given. The following definitions are all bodies that allow only an arrangement that getting these canonical, then, be viewed as being equipped with this arrangement.

A body is exactly then Euclidean if he

• An ordered Pythagorean body with exactly two square classes
• A Pythagorean body with exactly two square classes and,
• A formally real body with exactly two square classes
• A field of characteristic 0 with exactly two square classes and
• A formally real body that does not allow formally real, quadratic field extension or
• Ordered, its order ( number of his positive numbers ), a subgroup of index 2 in its multiplicative group and its characteristic 0

Is.

Properties

A Euclidean body

• Always has the characteristic 0,
• Always contains infinitely many elements,
• Is never complete algebraic,
• Is always formally real and Pythagorean,
• Contains every pure quadratic equation exactly two different solutions,
• May be disposed on just a way
• Is then completed exactly real if is algebraically closed,
• Has as its only Körperautomorphismus the identity map.

A worsening of the latter property: If a field extension and is a Euclidean and a formally real body, then there is exactly one embedding mapping.

And a consequence of the latter property: A field extension with a Euclidean field extension is Galois if and only about when.

In geometric applications Euclidean bodies are usually part of the real numbers and so arranged Archimedean. That this may not be necessary as shown by the example of the hyper- real numbers.

Examples and counter-examples

The main example of a Euclidean body forms, the field of real numbers.

In addition, are considered important examples:

• The real algebraic numbers (which are the real numbers in the algebraic closure of the field of rational numbers)
• The hyper- real numbers.

For every subset of that contains the amount of from " constructible with compass and straightedge " real numbers is a Euclidean body. This body is the smallest Euclidean body, as is contained in the subset, and for the smallest Euclidean body at all: every Euclidean body contains an isomorphic part to the body.

• Said smallest Euclidean body consists of exactly those real algebraic numbers, for which a square tower of field extensions exist, and that is for one. Necessary for the existence of the tower is that the extension degree of the field extension is a power of 2.
• The Euclidean body of constructible from a set with ruler and compass numbers consists exactly of those real and algebraic numbers, for which a tower exists, and the degree of over is then necessary for a power of 2

In all cases, the Euclidean body over infinite-dimensional vector spaces, that is, field extensions of infinite degree.

• Is an example of a formal real body is not Euclidean.
• The smallest Euclidean body is Euclidean, but not real completed because the zeros of the algebraic completion of all the level 3 and have therefore not be in, so can not be algebraically closed.

Euclidean planes and the Euclidean plane

Euclidean planes in the synthetic geometry satisfy all the axioms of the axiom groups I to IV in Hilbert's axiom system, but not always, the two continuity axioms that form the group V:

• V.1. ( Axiom of measuring or Archimedean axiom ). Are any lines AB and CD, so there is a number n such that the n-multiple cascading removal of the route also leads CD of A on the half line passing through B over the point B.
• V.2. ( Axiom of the (linear) completeness ) can Among the points of a straight line be added after receipt of the command and Kongruenzbeziehungen, no further points, and without existing under the previous elements relationships, the following from the axioms I-III basic properties of the linear arrangement and congruence or the axiom V.1 violated.

In Nonstandardmathematik (see Internal Set Theory ) can be transmitted, the Archimedean Axiom: instead a finite number of ablations, then the hyper finite number authorized in the Nonstandardversion in the inner volume. For these Euclidean planes then (with appropriate transfer of all other axioms which refer to infinite subsets of the plane or finite quantities counted with an indefinite number of terms ) is exactly the real and the hyper- real Euclidean plane a model. - In this hyper- real Euclidean plane, a regular polygon have a well-defined hyper finite number of corners. The standard synthetic geometry does not provide via the Euclidean body of the hyper real numbers this geometry.

In the standard geometry the axioms of continuity are replaced by axioms of the circle, to ensure that the constructions with ruler and compass never lead out of the coordinate area. Then meet exactly the plains Euclidean bodies, as described in this article, the new axiom system.

Another axiom system which describes these Euclidean planes is obtained when in addition to the axioms of a Pythagorean level should take the following arrangement Euclidean axiom:

The arrangement "between" straight on the plane must of course fulfill, which it follows that it is induced by one of the bodies in formally real Pythagorean always possible body configurations, the other arrangements required by arrays on a Pythagorean plane geometric properties.

An affine plane is exactly then Euclidean (in the sense of synthetic geometry), if it is a Pythagorean plane and satisfies (E). Each coordinate on a Euclidean plane body is by the arrangement of which is induced by the body in the only possible arrangement, and the (up to coordinates transformation ), the only possible to such an orthogonal Euclidean plane. Each Euclidean plane is isomorphic to such a coordinate plane on a Euclidean body.

Importance of the Euclidean axiom arrangement

To its form, Euclidean axiom arrangement only requires that the arrangement of the plane whose existence the axiom calls, be compatible with the orthogonality relation defined on the Pythagorean level. It is noteworthy that this " clearly obvious " compatibility requirement implies that in general only one arrangement of the plane is possible and that the plane is closed under constructions with ruler and compass.

• A "strong" arranged pappussche level is always isomorphic to a coordinate plane on a parent body. Such bodies always contain at least the two square classes. It can therefore be defined at the level always an orthogonality relation. (→ Präeuklidische level).
• The orthogonality must be designed for a Pythagorean plane so that every angle of the plane can be halved ( the layer must be free to move ), so special any right angles, from which the existence of squares and thus follows that the Orthogonalitätskonstante is equivalent to quadratic. (→ Präeuklidische level # squares).
• Especially under these conditions still exist infinitely many acceptable orthogonality: If you choose a fixed reference system, then delivers each number in the parent body as Orthogonalitätskonstante (relative to this coordinate system ) another orthogonality! Each of these orthogonality results in a Euclidean plane but, when used (E) to define the arrangement of the assembly at the same level.

Motivation

The examples of Euclidean body make it clear, so that the generalization of the real plane geometry is motivated: The Euclidean levels reflect what constructions are possible with certain provisions of the crowd. Figures, which can not be made ​​construct with ruler and compass, are simply not available from constructed Euclidean plane! While the real accounts of it were the largest is on algebraic body on which a Euclidean plane (in the sense of synthetic geometry) can be constructed which contains the " default lengths", referred to in the examples Euclidean bodies are the smallest body with these characteristics.

Archimedean Euclidean plane

A Euclidean plane is exactly then arranged Archimedean (in short: Archimedean ), that satisfies the axiom V.1 of measuring if their coordinates body an Archimedean (short for: Archimedean ordered ) body. This is obviously the case precisely when this Euclidean body is isomorphic to a sub- field of real numbers. In this case, there is - due to the particular arrangement of algebraic clearly - exactly one embedding and the body can always be identified with the " real " model.

A "small" and geometrically at best for counterexamples interesting model of a non- Archimedean Euclidean body is obtained when the rational function field in one variable over the rational numbers ( with the arrangement, etc.) analogous to that described above for construction within its algebraic statements real quadratic concludes.

Analytical Geometry in Euclidean planes

In analytic geometry, among other

Determined. In both aspects behaves the Euclidean plane over a Archimedean Euclidean body, which is here identified with his " real model ", substantially as the affine and Euclidean plane over because crucial here is the existence ( or nonexistence ) of eigenvalues ​​for - matrices with entries. Eigenvalues ​​which are the roots of a characteristic polynomial of degree 2 here, exist for such matrices exactly then, when they exist in! Each matrix with entries from that is real diagonalizable is also diagonalizable, it has a Jordan form over, then it is also about jordan between normal form similar to this.

Especially for quadratic forms and quadrics is significant that a symmetric matrix with entries from can be diagonalized by an orthogonal matrix with entries from. The eigenvalues ​​of this Euclidean normal form are either 0 or 1, or a square is equivalent to -1 therefore exist in the Euclidean plane over the same number of equivalence classes of affine quadrics as to the real coordinate plane. (→ Refer to the main axis transformation )

This agrees in general with only the two-dimensional affine space, ie in the plane.

Levels with radians

For a clear representation of a Euclidean body is seen in the following as part of the body, even if analogous constructions for non- Archimedean Euclidean levels and body are possible. The illustrated here introducing a radian means that not lead out on one level, from the constructions with ruler and compass, which is thus a Euclidean plane, an additional design tool " protractor " is introduced, with which it is possible arc lengths (angular extent ) on routes " handle ".

For this, the oriented Euclidean plane is identified by the number level. The orientation of and thus has the technical purpose that identifying with the correct direction of rotation takes place, so that rotations preserve the correct sign in the mathematically positive sense. A rotation of the plane about the point of origin may be represented by multiplication by a complex number:

For each rotation of the real plane around the origin corresponds to reversible clearly a number on the complex unit circle. The unit circle at the same time as a group of isomorphic to the group of rotation about the origin and provides for the rotation through the angle, and thus for each oriented angle ( oriented ) radians ( an oriented square in the normal sense ) the exception of the addition of multiple the full angle measure is unique.

The group of turns of the euclidean plane around the origin of the subgroup is isomorphic to

One defines: A surjective homomorphism

Of course at each coordinate plane over a part of the body of the rotation angle, and can be described by the real numerical values ​​. The point here is that each class of equal-length sections with the length in the Euclidean plane, which is, by the radian reversible clearly corresponds to a rotation of the plane and the addition of numbers, that is the concatenation removal of such systems with the composition of the associated rotations matches!

The existence of an arc measure is an additional axiom in synthetic Euclidean geometry levels, it is also referred to as a protractor axiom. Its validity is independent of the other axioms, the smallest Euclidean plane has no radians, nor the plane over the field of real algebraic numbers.

In an aligned, arranged Archimedean Euclidean plane with radian there are to each number exactly one radian, which has a circular number. This radians determines an oriented angle measure, ie for two rotations if and only if it is ( for "" ) or inverse to each other ( for " are ").

In the oriented plane are exactly the homomorphisms radians in terms of synthetic geometry, their channel count then.

In a Euclidean plane with radian

The second statement, which is sometimes referred to as angular pitch axiom implies the first. Both properties has not the smallest Euclidean plane, while the Euclidean plane over the real algebraic numbers both satisfied. So they are not sufficient conditions for the existence of a radian.

Sufficiently well for the existence of a radian, that the limitations of the trigonometric functions and the Euclidean body only pictures in. This observation can be a countable ( as a set ) Euclidean body design, the coordinate plane has a radians.