Pythagorean field

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:

• $ℝ$ and $ℂ$ 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

$$F=K_1\subseteq K_2\subseteq \mathrm{\cdots }\subseteq K_n=K$$

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