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:

•
$\mathbb{R}$ and $\u2102$ 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 $i=1,\mathrm{\dots},n1$. Take the compositum $L$ of the family $\mathcal{K}$ of all such $K$’s. Then $L={F}_{py}$.
Title  Pythagorean field 

Canonical name  PythagoreanField 
Date of creation  20130322 14:22:36 
Last modified on  20130322 14:22:36 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  11 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 12D15 
Defines  Pythagorean extension 
Defines  Pythagorean closure 