Pythagorean field

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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.


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