Pythagoras number

The Pythagoraszahl of a body is defined as the smallest, so that any finite sum of squares in already can write as the sum of squares.

Definition

Was for a body

The set of finite sums of squares that are equal to zero.

With

We denote the set of sums of squares in which at most have length. Obviously applies to everyone. It is unclear, however, whether always exists such that. As of Pythagoraszahl we denote the following size:

Where if and only if for all. It is always.

The Pythagoraszahl some number field

Further examples and evidence

If set non- real body, ( ie, ) can be estimated from the Pythagoraszahl by the step of:

Proof: See the first sentence ( Pythagoraszahl non- real body )

If a non - real body positive characteristic is a lemma that ( cf. the proof stage) applies from the Book of Squares AR Rajwade, after applies to any body.

This applies to all non - real field with positive characteristic that.

Quite accurately, you can be in the case where an odd Primpotenz is. The following applies:

Rate for all where is prime and.

Proof: See the first sentence ( Pythagoraszahl of bodies with a characteristic Primpotenz )

The Pythagoraszahl in field extensions of the rational numbers

Let be a finitely generated field extension over the rational numbers, should further the transcendence degree of over.

Using the Milnorschen conjecture ( cf. K-theory: Milnorvermutung ), which was proved by Vladimir Wojewodski, it can be shown that applies to everyone.

Because this estimate is sharp for.

For been previously shown. Presumably, however, applies even what then would be due to a sharp estimate.

A detailed presentation of the proof of can be found in the work over the Pythagoraszahl of function fields, see below