# Pythagorean field

A field in which any sum of two squares is a square. The **Pythagorean closure** of a field $K$ is the minimal Pythagorean subfield $K^\pi$ of the algebraic closure $\bar K$ containing $K$.

The **Pythagoras number** of a field $K$ is the smallest integer $d$, if it exists, such that every sum of squares in $K$ is already a sum of at most $d$ squares (or $\infty$ if no such number exists). A Pythagorean field has Pythagoras number equal to $1$. Lagrange's theorem implies that the Pythagoras number of the field of rational numbers is $4$. A finite field has Pythagoras number $1$ (in characteristic $2$) or $2$ (in characteristic $>2$). Every positive integer occurs as the Pythagoras number of some formally real field.

#### References

- Tsit Yuen Lam,
*Introduction to Quadratic Forms over Fields*, Graduate Studies in Mathematics**67**, American Mathematical Society (2005)**ISBN**0-8218-1095-2 Zbl 1068.11023 MR2104929 - A. R. Rajwade,
*Squares*, London Mathematical Society Lecture Note Series**171**Cambridge University Press (1993)**ISBN**0-521-42668-5 Zbl 0785.11022 - J.W. Milnor, D. Husemöller,
*Symmetric bilinear forms*, Ergebnisse der Mathematik und ihrer Grenzgebiete**73**, Springer-Verlag (1973)**ISBN**0-387-06009-X Zbl 0292.10016

**How to Cite This Entry:**

Pythagorean field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Pythagorean_field&oldid=54479