Pythagoras number

Not to be confused with Pythagoras's constant or Height of a field.

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 $p(\mathbb {R} )=1$.
• For a finite field of odd characteristic, not every element is a square, but all are the sum of two squares, 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.

• The Pythagoras number is related to the Stufe by $p(F)\leq s(F)+1$. If $F$ is not formally real then $s(F)\leq p(F)\leq s(F)+1$, and both cases are possible: for $F=\mathbb {C}$ we have $s=p=1$, whereas for $F=\mathbb {F} _{5}$ we have $s=1$, $p=2$.

• 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 $(s(F),p(F))=(s,p)$. For example,
  • quadratically closed fields and fields of characteristic 2 give $(s(F),p(F))=(1,1)$;
  • for primes $p\equiv 1\!\!\!\!{\pmod {4}}$, $\mathbb {F} _{p}$ and the p-adic field $\mathbb {Q} _{p}$ give $(1,2)$;
  • for primes $p\equiv 3\!\!\!\!{\pmod {4}}$, $\mathbb {F} _{p}$ 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 $h(F)=2s(F)$.

References

• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.

• Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.