Pythagorean field

In mathematics, a body referred to a set of elements ( " Numbers"), on which the four basic arithmetic operations according to certain rules apply. This body is referred to as Pythagorean if any additional (finite) sum of squares of the body is still a square.

This is not self-evident: A well-known from the school mathematics body is that of fractional numbers. Any sum or difference, product and quotient of each is in it always be determined. Since no rational square number is, this body is not Pythagorean.

Pythagorean body play an important role in synthetic geometry, there is often additionally required that -1 should not be a square number. They are then formally real Pythagorean body. - In the conventional view that 0 is not a square number, which is also used in this article, the additional property clear from the definition of the Pythagorean body. In these bodies, an arrangement is always possible. Präeuklidische a level above a formal real Pythagorean body in which the Orthogonalitätskonstante can be normalized to -1 is also referred to as a Pythagorean level. In such bisector planes can be constructed and it can introduce a notion of distance between points, which is based on the Pythagorean theorem of Euclidean planes. This is one of the reasons for the designation of " Pythagorean ".

A certain importance Pythagorean body and especially Pythagorean extensions to the question of solvability of Diophantine equations in elementary number theory.

Every Euclidean body is a formally real Pythagorean body. All these bodies always have the characteristic 0 and always contain infinitely many elements.

Definitions

A body is called Pythagorean body when one of the following equivalent conditions is true.

• The sum of two squares in K is a square number again.
• For each is a square number, ie.

For these formulations, it also follows that -1 is not a square number and thus no sum ​​of square numbers. Because then it would be would be the sum of squares of a square number, a contradiction because square numbers may not disappear.

Properties

A Pythagorean body as defined here is therefore always formally real. To emphasize this, the attribute is formally real added frequently, it then follows:

• The square-classes of -1 and 1 are different,
• The number -1 is not a square number,
• The characteristic of the field is 0

Different meanings

The occasionally used, weaker definition is obtained by the following characterizations: A body is called Pythagorean body ( in general form ) if its characteristic is 0, and additionally apply any of the following equivalent conditions:

• The sum of two square numbers is in back in,
• For each,
• The Pythagoraszahl of 1,
• Each Pythagorean expansion (see below) of not accord with.

An even weaker form, which is also found in the literature, also waived nor to the requirement that the characteristic should be 0. Even the characterizations mentioned in this section are four equivalent definitions of the attenuated concept.

Pythagorean expansion

A field extension of the form is called Pythagorean extension.

Strict - Pythagorean body

A body is called strictly - Pythagorean, when he is formally real and pythagorean formally real extension and each body is a Pythagorean body, provided that the field extension is square, so is their degree of expansion.

Euclidean body

A Pythagorean body is called Euclidean body when one of the following equivalent conditions is true:

• Each element of is either a perfect square or the negative of a square number, never both.
• The body contains exactly two square classes.

Both of these properties exacerbate the same time demanded by formal real bodies properties, even if " Pythagorean " is understood here in the broadest sense. So every body is a formally real Euclidean Pythagorean body with exactly two square classes.

Pythagorean level

In the synthetic geometry of an affine plane with orthogonality whose coordinates body is a formally real Pythagorean body and in a square ( the geometric figure! ) Exists, called the Pythagorean level. ( In this definition, the additional condition be " formally real " omitted, since the existence of squares implies that -1 not a square number).

Geometric applications

• The coordinates of a body präeuklidischen level that is freely movable ( in the cut for each pair of lines bisecting one exists), is a formally real Pythagorean body.
• Conversely, for a real formal Pythagorean body coordinate plane orthogonal with a floating präeuklidische level if the square is equivalent to Orthogonalitätskonstante -1.