| Version current |
Version 7 |
|
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.
|
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 $F$ $F$. A field $F$ is \emph{Pythagorean} if every Pythagorean extension of $F$ is $F$ itself.
|
|
|
| The following are equivalent: |
The following are equivalent: |
| \begin{enumerate} |
\begin{enumerate} |
| \item |
\item |
| $F$ is Pythagorean |
$F$ is Pythagorean |
| \item |
\item |
| Every sum of two squares in $F$ is a square |
Every sum of two squares in $F$ is a square |
| \item |
\item |
| Every sum of (finite number of) squares in $F$ is a square |
Every sum of (finite number of) squares in $F$ is a square |
| \end{enumerate} |
\end{enumerate} |
|
|
| \textbf{Examples:} |
\textbf{Examples:} |
| \begin{itemize} |
\begin{itemize} |
| \item |
\item |
| $\mathbb{R}$ and $\mathbb{C}$ are Pythagorean. |
$\mathbb{R}$ and $\mathbb{C}$ are Pythagorean. |
| \item |
\item |
| $\mathbb{Q}$ is not Pythagorean. |
$\mathbb{Q}$ is not Pythagorean. |
| \end{itemize} |
\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 |
\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 |
| $$F=K_1\subseteq K_2\subseteq \cdots \subseteq K_n=K$$ |
$$F=K_1\subseteq K_2\subseteq \cdots \subseteq K_n=K$$ |
| 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 $L=F_{py}$. |
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 $L=F_{py}$. |
