Pythagorean number

The Pythagoras number of a body is defined as the smallest , so that every finite sum of squares can be written as a sum of squares. ${\displaystyle F}$ ${\displaystyle p(F)\in \mathbb {N} }$ ${\displaystyle F}$ ${\displaystyle p(F)}$

definition

For a body be ${\displaystyle F}$

${\displaystyle \sum F^{2}:=\left\{\sum _{i=1}^{n}a_{i}^{2}\;{\Big |}\;n\in \mathbb {N} ,a_{1},\ldots ,a_{n}\in F{\text{ und }}\sum _{i=1}^{n}a_{i}^{2}\neq 0\right\}}$

the set of finite sums of squares that are not zero.

With

${\displaystyle \sum ^{k}F^{2}:=\left\{\sum _{i=1}^{n}a_{i}^{2}\;{\Big |}\;n\leq k,a_{1},\ldots ,a_{n}\in F{\text{ und }}\sum _{i=1}^{n}a_{i}^{2}\neq 0\right\}}$

we denote the set of sums of squares in which are at most length . Obviously applies to everyone . It is unclear, however, whether there always exists such that . As Pythagoras number of we call the following size: ${\displaystyle F}$ ${\displaystyle k}$ ${\displaystyle \textstyle \sum ^{k}F^{2}\subseteq \sum F^{2}}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle \textstyle \sum ^{k}F^{2}=\sum F^{2}}$ ${\displaystyle F}$

${\displaystyle p(F):=\min \left\{k\in \mathbb {N} \cup \{\infty \}\;{\Big |}\;\sum ^{k}F^{2}=\sum F^{2}\right\}}$

where if and only if applies to all . It is always . ${\displaystyle p(F)=\infty }$ ${\displaystyle \textstyle \sum ^{k}F^{2}\varsubsetneq \sum F^{2}}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle p(F)\geq 1}$

The Pythagorean number of some number fields

1. According to the Pythagorean theorem there is for one such that . So the Pythagoras number is the real number . In other words: You can extract the root of every sum of squares . It is likely that the Pythagorean number derives its name from this consideration. ${\displaystyle a_{1},\ldots ,a_{n}\in \mathbb {R} }$ ${\displaystyle b\in \mathbb {R} }$ ${\displaystyle a_{1}^{2}+\ldots +a_{n}^{2}=b^{2}}$ ${\displaystyle p(\mathbb {R} )=1}$ ${\displaystyle \mathbb {R} }$
2. The Pythagorean number of complex numbers . ${\displaystyle p(\mathbb {C} )=1}$
3. According to the Euler-Lagrange theorem , the Pythagorean number of the rational numbers , i.e. H. every sum of squares of rational numbers can be written as a sum of at most four squares. ${\displaystyle p(\mathbb {Q} )=4}$

More examples and evidence

Theorem If is a non-real field , (that is ,) the Pythagoras number of can be estimated by the level of : ${\displaystyle F}$ ${\displaystyle -1\in \sum F^{2}}$ ${\displaystyle F}$ ${\displaystyle s(F)}$ ${\displaystyle F}$

${\displaystyle s(F)\leq p(F)\leq s(F)+1}$

Proof: See theorem (Pythagorean number of non-real fields)

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If a non-real body having a positive characteristic is a lemma from the book Squares of AR Rajwade, after for any body with true that (cf. the proof. Level ). ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle {\text{char}}(F)>0}$ ${\displaystyle s(F)\leq 2}$

This means that for all non-real bodies with positive characteristics, that . ${\displaystyle p(F)\leq 3}$

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One can become very exact in the case where there is an odd prime power. The following applies: ${\displaystyle F=\mathbb {F} _{q}}$ ${\displaystyle q}$

Sentence for all where prime and is. ${\displaystyle p(\mathbb {F} _{q})=2}$ ${\displaystyle q=p^{n}}$ ${\displaystyle p>2}$ ${\displaystyle n>0}$

Proof: See theorem (Pythagoras number of fields with the characteristic of a prime power)

The Pythagoras number for the field extensions of the rational numbers

Let there be a finitely generated body extension over the rational numbers , let the degree of transcendence of over . ${\displaystyle F/\mathbb {Q} }$ ${\displaystyle d={\text{trdeg}}(F)}$ ${\displaystyle F}$ ${\displaystyle \mathbb {Q} }$

Using the Milnor's conjecture (see K-theory: Milnor 's conjecture ), which was proven by Vladimir Wojewodski , it can be shown that applies to everyone . ${\displaystyle p(F)\leq 2^{d+2}}$ ${\displaystyle d}$

Because of this, this estimate is keen for . ${\displaystyle p(\mathbb {Q} )=4}$ ${\displaystyle d=0}$

For has been shown so far . Presumably, however , what would then be a sharp estimate also applies . ${\displaystyle d=1}$ ${\displaystyle p(F)\leq 6}$ ${\displaystyle p(F)\leq 5}$ ${\displaystyle p(\mathbb {Q} (t))=5}$

A detailed presentation of the proof of can be found in the work On the Pythagorean number of function fields , see. u. ${\displaystyle p(F)\leq 2^{d+2}}$

See also

• Pythagorean body

Web links

Wikibooks: Some Evidence on the Pythagorean Number  - Study and Teaching Materials

• About the Pythagoras number of function fields

Individual evidence

1. ^ Bröcker L., About the Pythagoras number of a body , Archive of Mathematics, Birkhäuser Basel, Volume 31, Number 1, December 1978, pp. 133-136
2. ^ AR Rajwade, Squares , Cambridge University Press, 1993
3. ↑ Florian Pop, previously unpublished article
4. ↑ Y. Pourchet, Sur la representation en somme de carres the polynomial a une Indéterminée sur un corps de nombres algebraiques , Acta Arith. 19, 1971