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'Euclidean field'
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| Title of object: |
Euclidean field |
| Canonical Name: |
EuclideanField |
| Type: |
Definition |
| Created on: |
2004-05-21 19:59:41 |
| Modified on: |
2007-06-13 05:54:40 |
| Classification: |
msc:12D15 |
| Defines: |
straightedge and compass operation, compass and straightedge operation, constructible number, constructible from, constructible, Euclidean, field of real constructible numbers |
Preamble:
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Content:
\PMlinkescapeword{close}
\PMlinkescapeword{constructible}
\PMlinkescapeword{Euclidean}
\PMlinkescapeword{length}
\PMlinkescapeword{level}
\PMlinkescapeword{measure}
\PMlinkescapeword{open}
\PMlinkescapetext{This page is under construction.}
An ordered field $F$ is \emph{Euclidean} if every non-negative element $a$ ($a\geq0$) is a square in $F$ (there exists $b\in F$ such that $b^2=a$).
For example, $\mathbb{R}$ is Euclidean. On the other hand, $\mathbb{Q}$ is not Euclidean because 2 is not a square in $\mathbb{Q}$ (\PMlinkname{i.e.}{Ie}, $\pm\sqrt{2}\notin \mathbb{Q}$). Also, $\mathbb{C}$ is not a Euclidean field because \PMlinkname{$\mathbb{C}$ is not an ordered field}{MathbbCIsNotAnOrderedField}.
A Euclidean field is an ordered Pythagorean field.
There are ordered fields that are Pythagorean but not Euclidean.
The smallest subfield $\mathbb{E}$ of $\mathbb{R}$ over $\mathbb{Q}$ such that $\mathbb{E}$ is Euclidean is called the \emph{field of real constructible numbers}. A element of $\mathbb{E}$ is called a \emph{constructible number}. These numbers can be ``constructed'' by a process that will be described shortly. First, note that $\mathbb{E}$ has the following properties:
\begin{enumerate}
\item $0,1\in\mathbb{E}$;
\item If $a,b\in\mathbb{E}$, then so are $a\pm b$, $ab$, and $a/b\in\mathbb{E}$, the last of which is meaningful only when $b\not=0$;
\item If $r\in\mathbb{E}$ and $r>0$, then $\sqrt{r}\in\mathbb{E}$.
\end{enumerate}
Conversely, let us start with a subset $S$ of $\mathbb{R}$ such that $S$ contains a non-zero real number. Call any of the binary operations in condition 2 above, and the square root unary operation in condition 3 a \emph{ruler and compass operation}. (Note that, in mathematics, phrases such as ``ruler and compass'' and ``compass and straightedge'' \PMlinkescapetext{mean} the same thing.) Call a real number \emph{constructible from} $S$ if it can be obtained from elements of $S$ by a finite sequence of ruler and compass operations. Note that $1\in S$. If $S^{\prime}$ is the set of numbers constructible from $S$ using only the binary ruler and compass operations (those in condition 2), then $S^{\prime}$ is a subfield of $\mathbb{R}$, and is the smallest field containing $S$. Next, denote $\hat{S}$ the set of all constructible numbers from $S$. It is not hard to see that $\hat{S}$ is also a subfield of $\mathbb{R}$, but an extension of $S^{\prime}$. Furthermore, it is not hard to show that $\hat{S}$ is Euclidean. The general process (algorithm) of \PMlinkescapeword{generating} elements in $\hat{S}$ from elements in $S$ using finite sequences of ruler and compass operations is called a \emph{ruler and compass construction}. These are so called because, given two points, one of which is 0, the other of which is a non-zero real number in $S$, one can use a ruler and compass to construct these elements of $\hat{S}$.
If $S=\lbrace 1\rbrace$ (or any rational number), we see that $\hat{S}=\mathbb{E}$ is \emph{the} field of constructible numbers. |
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