Pythagorean body

In mathematics , a body describes a set of elements ("numbers") to which the four basic arithmetic operations can be applied according to certain rules. This field is called Pythagorean if, in addition, every (finite) sum of square numbers of the field is still a square number.

This cannot be taken for granted: A body known from school mathematics is that of fractions . Any sum or difference, any product and any quotient can always be determined. Since is not a rational square number, this body is not Pythagorean. ${\displaystyle 1^{2}+1^{2}=1+1=2}$

Pythagorean solids play an important role in synthetic geometry , where it is often required that −1 should not be a square number. They are then formally real Pythagorean solids. - With the usual view that 0 is not a square number, which is also used in this article, the additional property results from the definition of the Pythagorean field. An arrangement is always possible for these bodies . A pre-Euclidean level over a formally real Pythagorean field, in which the orthogonality constant can be normalized to −1, is also known as a Pythagorean level . In such planes, bisectors can be constructed and a notion of distance between points can be introduced, which is based on the Pythagorean theorem of Euclidean planes. This is one of the reasons for the term "Pythagorean".

Pythagorean solids and especially Pythagorean extensions have a certain meaning for the question of the solvability of Diophantine equations in elementary number theory .

Every Euclidean solid is a formally real Pythagorean solid. All these bodies always have the characteristic 0 and always contain an infinite number of elements.

Definitions

A body is called a Pythagorean body if one of the following equivalent conditions is met. ${\displaystyle (K,+,\cdot )}$

• The sum of two square numbers in K is again a square number.
• For each is a square number, so . ${\displaystyle a\in K}$ ${\displaystyle 1+a^{2}}$ ${\displaystyle 1+a^{2}\in Q_{1}={K^{*}}^{2}=\lbrace b^{2}:\;b\in K\setminus \lbrace 0\rbrace \rbrace }$

From these formulations it follows at the same time that −1 is not a square number and therefore also not a sum of square numbers. Because if , then as a sum of square numbers there would also be a square number, a contradiction, because square numbers must not disappear. ${\displaystyle -1=c^{2},c\in K^{*}}$ ${\displaystyle 0=c^{2}+1^{2}}$

properties

A Pythagorean solid as defined here is always formally real. To emphasize this, the attribute is often added formally real , from which it then follows:

• The square classes of −1 and 1 are different,
• the number −1 is not a square number,
• the characteristic of the body is 0.

Different meanings

The sometimes used, weaker definition is obtained by the following characterizations: A body is called a Pythagorean body (in more general form) if its characteristic is 0 and one of the following equivalent conditions also applies: ${\displaystyle (K,+,\cdot )}$

• the sum of two square numbers in is again in , ${\displaystyle K}$ ${\displaystyle K^{2}=\lbrace k^{2}|k\in K\rbrace }$
• each is , ${\displaystyle a\in K}$ ${\displaystyle 1+a^{2}\in K^{2}}$
• the Pythagorean number of is 1, ${\displaystyle K}$
• every Pythagorean extension (see below) of matches with . ${\displaystyle K}$ ${\displaystyle K}$

An even weaker form, which also appears in the literature, also dispenses with the requirement that the characteristic should be 0. Even then, the four characterizations mentioned in this section are equivalent definitions of the weakened term.

Pythagorean extension

A body extension of the form is called a Pythagorean extension . ${\displaystyle K({\sqrt {1+a^{2}}}),\!\,\quad (a\in K^{*})}$

Strict Pythagorean body

A body is called strictly Pythagorean if it is formally real and Pythagorean and every formally real expansion body is a Pythagorean body, provided that the body expansion is square, i.e. its degree of expansion . ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle K<L}$ ${\displaystyle (L:K)=2}$

Euclidean body

A Pythagorean field is called a Euclidean field if one of the following equivalent conditions is true: ${\displaystyle K}$

• Each element of is either a square number or the negative of a square number, never both. ${\displaystyle K\setminus \lbrace 0\rbrace }$
• The body contains exactly the two square classes and . ${\displaystyle Q_{1}}$ ${\displaystyle Q_{-1}}$

Both of these properties also intensify the properties required of formally real bodies, even if “Pythagorean” is understood here in the broadest sense. So every Euclidean field is a formally real Pythagorean field with exactly 2 square classes.

Pythagorean plain

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

Geometric applications

• The coordinate body of a pre-Euclidean plane that is freely movable (in which there is an angle bisector for each intersecting pair of lines ) is a formally real Pythagorean body.
• Conversely, for a formally real Pythagorean solid, the coordinate plane with orthogonality is a freely movable pre-Euclidean plane if the orthogonality constant is quadratically equivalent to −1. ${\displaystyle K}$ ${\displaystyle K^{2}}$

literature

• L. Bröcker, On a Class of Pythagorean Solids , Archives of Mathematics, Volume 23, Number 1, December 1972
• Wendelin Degen, Lothar Profke: Fundamentals of affine and Euclidean geometry. Teubner, Stuttgart 1976, ISBN 3-519-02751-8 .

Web links

• Pythagorean Field at PlanetMath (English)

Individual evidence

1. ↑ For the notation: In the current Duden - The great dictionary of the German language in ten volumes - ISBN 3-411-70360-1 , the adjective "Pythagorean" given in this notation and "Pythagor spelling ä isch" called Austrian special form. In the German-language mathematical specialist literature, both spellings occur without any related difference in meaning.
2. ↑ In this article, according to Degen (1976), a square number is always understood to be a body element , so the 0 is excluded. ${\displaystyle k^{2}\neq 0;k\in K}$
3. ↑ Degen (1976), p. 146
4. ↑ Eric W. Weisstein: Pythagorean Field. From MathWorld - A Wolfram Web Resource.
5. ↑ Eric W. Weisstein: Pythagorean Extension. From MathWorld - A Wolfram Web Resource.
6. ↑ Brocker (1972), pp. 405-407