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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: |
2007-06-07 14:45:54 |
| 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 |
Revision comment (for changes between this and next version):
| killing links (previewer did not work to show me extraneous links!) |
Preamble:
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Content:
\PMlinkescapeword{constructible}
\PMlinkescapeword{Euclidean}
\PMlinkescapeword{length}
\PMlinkescapeword{measure}
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 $b^2=a$).
A Euclidean field is an ordered Pythagorean field. 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}. 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.
Ruler and compass constructions are of historical significance. The ancient Greeks are the most well-known civilization for investigating these constructions on an elementary level. It should be pointed out that the compasses that they used were \emph{collapsible}. That is, you could open the compass and draw an arc, but immediately after you removed a point of the compass from the plane were you drew the arc, the compass would \PMlinkescapetext{close} completely. It turns out that whether a collapsible compass or a modern-day compass is used to perform these constructions makes no difference. More precisely put, the field of constructible numbers when one uses a collapsible compass and the field of constructible numbers when one uses a modern-day compass are equal.
Outside of mathematics, a ruler is used to measure any \PMlinkname{length}{BasicLength} desired. This is \emph{not} the \PMlinkescapetext{type} of ruler that is being used in these ruler and compass constructions. For this reason, the tool with which line segments are constructed is sometimes called a straightedge instead of a ruler. This makes it clear that this tool cannot measure lengths that are \PMlinkescapetext{independent} of the distance from 0 to 1. Other lengths are only obtainable from a finite sequence of compass and straightedge operations. |
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