Pythagoras number

In mathematics, the Pythagoras number or reduced height of a field describes the structure of the set of squares in the field. The Pythagoras number $p(K)$ of a field $K$ is the smallest positive integer $p$ such that every sum of squares in $K$ is a sum of $p$ squares.

A Pythagorean field is a field with Pythagoras number 1: that is, every sum of squares is already a square.

Examples

• Every non-negative real number is a square, so $p(ℝ)=1$.
• For a finite field of odd characteristic, not every element is a square, but all are the sum of two squares,Template:Sfnp so $p=2$.
• By Lagrange's four-square theorem, every positive rational number is a sum of four squares, and not all are sums of three squares, so $p(\mathbb{Q})=4$.

Properties

• Every positive integer occurs as the Pythagoras number of some formally real field.Template:Sfnp

• The Pythagoras number is related to the Stufe by $p(F)\le s(F)+1$.Template:Sfnp If $F$ is not formally real then $s(F)\le p(F)\le s(F)+1$,Template:Sfnp and both cases are possible: for $F=ℂ$ we have $s=p=1$, whereas for $F={𝔽}_5$ we have $s=1$, $p=2$.Template:Sfnp

• As a consequence, the Pythagoras number of a non-formally-real field is either a power of 2, or 1 more than a power of 2. All such cases occur: i.e., for each pair $(s,p)$ of the form $(2^k,2^k)$ or $(2^k,2^k+1)$, there exists a field $F$ such that $(s(F),p(F))=(s,p)$.Template:Sfnp For example,
  • quadratically closed fields and fields of characteristic 2 give $(s(F),p(F))=(1,1)$;
  • for primes $p\equiv 1(\mathrm{mod}4)$, ${𝔽}_p$ and the p-adic field ${\mathbb{Q}}_p$ give $(1,2)$;
  • for primes $p\equiv 3(\mathrm{mod}4)$, ${𝔽}_p$ gives $(2,2)$ and ${\mathbb{Q}}_p$ gives $(2,3)$;
  • ${\mathbb{Q}}_2$ gives $(4,4)$, and
  • the function field ${\mathbb{Q}}_2(t)$ gives $(4,5)$.

• The Pythagoras number is related to the height of a field $F$: if $F$ is formally real then $h(F)$ is the smallest power of 2 which is not less than $p(F)$; if $F$ is not formally real then $h(F)=2s(F)$.Template:Sfnp

References

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