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