Euclidean body

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A Euclidean field is a field (in the sense of algebra ) that is an ordered field and in which every nonnegative element has a square root .

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

Euclidean bodies play an important role in synthetic geometry : the coordinate body of a Euclidean plane is always a Euclidean body, and a Euclidean plane can always be built on these bodies. The term "Euclidean plane" is somewhat more general than in the usual geometry, where the Euclidean plane is defined by Hilbert's system of axioms of Euclidean geometry in such a way that it is necessarily an affine plane above the special Euclidean field of real numbers - one to Hilbert's System equivalent formulation in the language of linear algebra reads: A Euclidean plane is an affine space whose vector space of displacements is a two-dimensional Euclidean vector space , i.e. isomorphic to with a scalar product . The Euclidean levels of synthetic geometry are closely related to classical questions of constructibility . These questions give rise to additional axioms, such as the protractor axiom , which requires the existence of a radian measure, and the angle division axiom , which cannot be fulfilled in all Euclidean planes.

If one replaces the real numbers as a coordinate area in the analytically formulated two- or three-dimensional, real Euclidean geometry by any Archimedean Euclidean body, then models for non-Euclidean geometries can be constructed within this geometry , which instead of the axioms of continuity (axiom group V in Hilbert's system of axioms ) fulfill the somewhat weaker axioms of the circle . In these models of absolute geometry , all constructions can still be carried out with compasses and rulers.

Euclidean solids have a certain importance as counterexamples in the theory of body extensions and Galois theory , as well as in transcendence studies in number theory .

Euclidean bodies and planes bear their names in honor of the ancient mathematician Euclid of Alexandria, both of which owe their name to his axiomatic structure of the Euclidean geometry in his work " The Elements ", which is still named after him today . → The term “ Euclidean ring ” from the theory of divisibility into commutative rings has no closer context to the terms described in this article than that it is also named after Euclid, namely after the Euclidean algorithm he describes .

Alternative definitions

A Pythagorean field , i.e. a field in which every sum of squares is again a square, is a Euclidean field if and only if it contains exactly the two square classes and . Although this purely algebraic definition does not yet give an arrangement, there is only one possibility in such Pythagorean bodies to make them an ordered body and that is through the definition

if and only if is a square (i.e. an element of ).

This means that this “canonical arrangement” can be seen as given by the algebraic structure. In the following definitions all bodies which only allow one arrangement, which is then always this canonical, are to be regarded as so arranged.

A body is Euclidean if and only if it

  • an ordered Pythagorean solid with exactly two square classes,
  • a Pythagorean field with exactly two square classes and ,
  • a formally real field with exactly two square classes,
  • a field of characteristic 0 with exactly two square classes and
  • a formally real body that does not allow a formally real, square body extension or
  • ordered, its order (set of its positive numbers), a subgroup of the index 2 in its multiplicative group and its characteristic 0



A Euclidean body

  • always has the characteristic 0,
  • always contains an infinite number of elements,
  • is never algebraically closed ,
  • is always formally real and Pythagorean,
  • contains exactly two different solutions for each purely square equation ,
  • can be arranged in exactly one way,
  • is real closed if and only if is algebraically closed,
  • The only body automorphism has the identical mapping (is a rigid body ).

A tightening of the last-mentioned property: If it is a body extension and is a Euclidean and a formally real body, then there is exactly one embedding of nach .

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

In geometric applications, Euclidean solids are mostly part of the real numbers and are therefore arranged in an Archimedean way . The example of hyper-real numbers shows that this does not have to be the case .

Examples and counterexamples

The main example of a Euclidean field is the field of real numbers .

Other important examples are:

For each subset of , which includes the amount of out is " constructible with ruler and compass " real numbers a Euclidean body. This body is the smallest Euclidean body that is contained as a subset, and for the smallest Euclidean body at all: Every Euclidean body contains a partial body that is too isomorphic.

  • The mentioned smallest Euclidean field consists of exactly those real algebraic numbers for which a tower of square field extensions exists such that and for is. It is necessary for the existence of the tower that the degree of expansion of the body is a power of 2 .
  • The Euclidean field of numbers that can be constructed from a set of compasses and ruler consists of precisely those real and over algebraic numbers for which a corresponding tower over exists, and the degree of over is then necessarily a power of 2.
The easiest way to understand geometrically is that the quantities described are actually solids, i.e. that with 2 elements with the specified property, their sum and product etc. also have the required property. → See constructible polygon .

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

  • is an example of a formally real field that is not Euclidean.
  • The smallest Euclidean field is Euclidean, but not real closed, because the zeros of in the algebraic closure of all have degree 3 above and are therefore not in , so it can not be algebraically closed.

Euclidean planes and the Euclidean plane

to the Archimedean axiom

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

  • V.1. (Axiom of Measurement or Archimedes' Axiom ) . If AB and CD are any lines, then there is a number n such that the n-times successive removal of the line CD leads from A to the half-ray going through B beyond point B.
  • V.2. (Axiom of (linear) completeness) No further points can be added to the points of a straight line, if their arrangement and congruence relationships are preserved, without the relationships existing under the previous elements, the basic properties of the linear ones following from axioms I-III Arrangement and congruence or axiom V.1 are violated.

In nonstandard mathematics (see Internal Set Theory ), the Archimedean axiom can be carried over: Instead of a finite number of reductions, the hyperfinite number is then allowed in the inner set in the nonstandard version . For these Euclidean planes (with the appropriate transfer of all other axioms that refer to infinite subsets of the plane or finite counts with an indefinite number) exactly the real and the hyper-real Euclidean plane are a model. - In this hyper-real Euclidean plane a regular polygon can also have a well-determined hyper-finite number of vertices. The synthetic standard geometry does not provide this geometry even over the Euclidean field of the hyper-real numbers .

In standard geometry, the axioms of continuity are replaced by axioms of compasses, which ensure that the constructions with compasses and ruler never lead out of the coordinate range. Then exactly the planes above Euclidean solids as described in this article satisfy the new system of axioms.

to the Euclidean axiom of arrangement

Another system of axioms describing these Euclidean planes is obtained if, in addition to the axioms of a Pythagorean plane, the following Euclidean arrangement axiom is added:

(E) There is an arrangement “between”, so that of three different collinear points, lies between and exactly when a right-angled triangle with the height base point can be built over the line .

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

An affine plane is Euclidean (in the sense of synthetic geometry) if and only if it is a Pythagorean plane and satisfies (E). Every coordinate plane over a Euclidean body becomes such a Euclidean plane through the arrangement induced by the only possible body arrangement and through the only possible orthogonality (except for coordinate transformation) . Every Euclidean plane is isomorphic to such a coordinate plane over a Euclidean body.

Meaning of the Euclidean axiom of arrangement

According to its form, the Euclidean axiom of arrangement only requires that the arrangement of the plane, the existence of which the axiom requires, is compatible with the orthogonality relation defined on the Pythagorean plane . It is noteworthy that this “clearly obvious” compatibility requirement implies that only one arrangement of the plane is possible and that the plane is closed under constructions with compasses and ruler.

  • A "strongly" arranged, Pappusian plane is always isomorphic to a coordinate plane over an ordered body. Such solids always contain at least the two square classes and . Therefore, an orthogonality relation can always be defined on the plane. (→ Pre-Euclidean level ).
  • The orthogonality must be such for a Pythagorean level that each angle of the plane can be halved (the layer must move freely to be), that special any right angles, from which the existence of squares follows, thus, that the Orthogonalitätskonstante to square equivalent is . (→ Pre-Euclidean plane # squares ).
  • Precisely under these prerequisites there are still an infinite number of compatible orthogonality relations: If one chooses a fixed reference system , then every number in the ordered body delivers a different orthogonality relation as an orthogonality constant (based on this coordinate system)! However, each of these orthogonality relations leads in a Euclidean plane, if one uses (E) to define the arrangement, to the same arrangement of the plane.


The examples for Euclidean solids make it clear what motivates the generalization of real plane geometry: The Euclidean planes reflect which constructions are possible with certain specifications of the set . Figures that can not be constructed from with compasses and ruler simply do not exist in the Euclidean plane constructed from ! While the real closure of is, so to speak, the largest over algebraic bodies on which a Euclidean level (in the sense of synthetic geometry) can be built that contains the “default lengths”, the Euclidean bodies named in the examples are the smallest bodies with these properties .

Archimedean Euclidean Plane

A Euclidean plane is arranged Archimedean (short: Archimedean), i.e. it fulfills the axiom V.1 of measurement, if its coordinate body is an Archimedean (short for: Archimedean ordered) body. This is obviously the case if and only if this Euclidean field is isomorphic to a subfield of the real numbers. In this case there is - due to the algebraically clearly determined arrangement of - exactly one embedding and the body can always be identified with the "real model" .

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

Analytical Geometry on Euclidean Levels

In analytical geometry , among other things

  1. Normal forms for the affine self-mapping of an affine space, especially the affinities in this space,
  2. Normal shapes for square shapes and their associated quadrics

certainly. In both aspects, the Euclidean plane behaves over an Archimedean Euclidean body , which is identified here with its "real model", essentially like the affine or Euclidean plane over , because the decisive factor here is the existence (or non-existence) of eigenvalues for - Matrices with entries . Eigenvalues, which are the zeros of a characteristic polynomial of degree 2 , exist for such matrices in if and only if they exist in! Every matrix with entries that can actually be diagonalized is also diagonalizable via , if it has a Jordanian normal form via , then it is also similar to this Jordanian normal form via via .

Especially for square shapes and quadrics it is important that a symmetrical matrix with entries from can be diagonalized by an orthogonal matrix with entries from . The eigenvalues ​​of this Euclidean normal form are then either 0 or quadratically equivalent to −1 or 1. Therefore there are as many affine equivalence classes of quadrics in the Euclidean plane as there are over the real coordinate plane . (→ see principal axis transformation )

In general, this only agrees in two- dimensional affine space , i.e. in the plane.

Planes with radians

For a clear representation, a Euclidean body is understood in the following as a partial body of , even if analogous constructions are also possible for non-Archimedean Euclidean planes and bodies. The introduction of a radian measure shown here means that an additional construction tool "protractor" is introduced on a plane from which constructions with compasses and ruler do not lead, which is a Euclidean plane, with which it is possible to measure arc lengths (angular dimensions) "Process" routes.

For this, the oriented Euclidean plane is over the number plane identified. The orientation of and thus has the technical purpose of identifying with in the correct direction of rotation, so that rotations in the mathematically positive sense are given the correct sign . A rotation of the point plane around the origin can then be represented by multiplying it with a complex number :


Every rotation of the real plane around the origin corresponds to a number on the complex unit circle . This unit circle is at the same time as a subgroup of isomorphic to the group of rotations around the origin and provides for the rotation around the angle and thus for each oriented angle an (oriented) radian measure (an oriented angle measure in the usual sense) , apart from the addition of multiples of the full angle dimension is unambiguous.

The group of rotations of the Euclidean plane about the origin is isomorphic to the subgroup

to define the radian measure

One defines: A surjective homomorphism

with the property that an element exists, so that the real part maps the interval bijectively and strictly monotonically falling on the interval and is, a radian measure on the Euclidean plane is called. The number is called the number of circles in radians.

Of course, the rotations and angles of every coordinate plane over a part of the body can be described by real dimensions. The decisive factor here is that each class has lines of equal length with the length in the Euclidean plane, whereby , reversible through the radian measure, clearly corresponds to a rotation of the plane and the addition of numbers, i.e. the successive subtraction of such lines with the composition of the associated Rotations match!

The existence of a radian measure is an additional axiom in synthetic geometry for Euclidean planes, it is also called the protractor axiom . Its validity is independent of the other axioms, the smallest Euclidean plane has no radian measure, nor does the plane above the field of real algebraic numbers.

In an oriented, Archimedean arranged Euclidean plane with radian measure, there is exactly one radian measure for every number , which is a circular number. This radian measure determines an oriented angle measure , that is, for two rotations is exactly when they are equal (for " ") or inversely to one another (for " ").

In the oriented plane the homomorphisms are precisely radians in the sense of synthetic geometry, their number is then .

In a Euclidean plane with radians

  1. there is a regular corner for every natural number ,
  2. there is a rotation with every rotation and every natural number .

The second proposition, sometimes referred to as the axiom of angular division , implies the first. The smallest Euclidean plane does not have both properties, while the Euclidean plane above the real algebraic numbers fulfills both. So they are not sufficient conditions for the existence of a radian measure.

It is sufficient for the existence of a radian that the constraints on the trigonometric functions and on the Euclidean body only have images in . With this observation, a countable Euclidean body can be constructed (as a set) whose coordinate plane has a radian measure.


Individual evidence

  1. Degen, Profke: Fundamentals of affine and Euclidean geometry. 1976, p. 149.
  2. Degen, Profke: Fundamentals of affine and Euclidean geometry. 1976, p. 149 ff.
  3. ^ Freudenthal: Mathematics as an educational task. Volume 1. 1973, p. 292 ff.
  4. ^ Theodor Schneider: Transcendence studies of periodic functions I. Transcendence of potencies. In: Journal for pure and applied mathematics. Vol. 172, H. 2, 1935, ISSN  0075-4102 , pp. 65-69, digitized .
  5. Degen, Profke: Fundamentals of affine and Euclidean geometry. 1976, p. 173 f.