Let $F$ be a field. A field extension $K$ of $F$ is called a Pythagorean extension if $K=F(\sqrt{1+\alpha^{2}})$ for some $\alpha$ in $F$, where $\sqrt{1+\alpha^{2}}$ denotes a root of the polynomial $x^{2}-(1+\alpha^{2})$ in the algebraic closure $\overline{F}$ of $F$. A field $F$ is Pythagorean if every Pythagorean extension of $F$ is $F$ itself.

The following are equivalent:

1. 1.

   $F$ is Pythagorean

2. 2.

   Every sum of two squares in $F$ is a square

3. 3.

   Every sum of (finite number of) squares in $F$ is a square

Examples:

• $\mathbb{R}$ and $\mathbb{C}$ are Pythagorean.

• $\mathbb{Q}$ is not Pythagorean.

Remark. Every field is contained in some Pythagorean field. The smallest Pythagorean field over a field $F$ is called the Pythagorean closure of $F$, and is written $F_{{py}}$. Given a field $F$, one way to construct its Pythagorean closure is as follows: let $K$ be an extension over $F$ such that there is a tower

of fields with $K_{{i+1}}=K_{i}(\sqrt{1+\alpha_{i}^{2}})$ for some $\alpha_{i}\in K_{i}$, where $i=1,\ldots,n-1$. Take the compositum $L$ of the family $\mathcal{K}$ of all such $K$’s. Then $L=F_{{py}}$.