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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 05:37:44 |
| Classification: |
msc:12D15 |
| Defines: |
ruler and compass construction, straightedge and compass construction, compass and straightedge construction, ruler and compass operation, straightedge and compass operation, compass and straightedge operation, constructible number, constructible from, constructible, Euclidean, field of real constructible numbers, collapsible compass |
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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