PythagoreanField 2004-05-21 19:30:38 2007-06-07 20:01:33 Definition Pythagorean extension Pythagorean closure % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $F$ be a field. A field extension $K$ of $F$ is called a \emph{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 \emph{Pythagorean} if every Pythagorean extension of $F$ is $F$ itself. The following are equivalent: \begin{enumerate} \item $F$ is Pythagorean \item Every sum of two squares in $F$ is a square \item Every sum of (finite number of) squares in $F$ is a square \end{enumerate} \textbf{Examples:} \begin{itemize} \item $\mathbb{R}$ and $\mathbb{C}$ are Pythagorean. \item $\mathbb{Q}$ is not Pythagorean. \end{itemize} \textbf{Remark}. Every field is contained in some Pythagorean field. The smallest Pythagorean field over a field $F$ is called the \emph{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 $L=F_{py}$.