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. $F$ is Pythagorean
2. Every sum of two squares in $F$ is a square
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 $$F=K_1\subseteq K_2\subseteq \cdots \subseteq K_n=K$$ 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 $L=F_{py}$