PlanetMath: constructible numbers

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constructible numbers

(Definition)

The smallest subfield $\mathbb{E}$ of $\mathbb{R}$ over $\mathbb{Q}$ such that $\mathbb{E}$ is Euclidean is called the field of real constructible numbers. First, note that $\mathbb{E}$ has the following properties:

$0,1\in\mathbb{E}$ ;
If $a,b\in\mathbb{E}$ , then also $a\pm b$ , , and $a/b\in\mathbb{E}$ , the last of which is meaningful only when $b\not=0$ ;
If $r\in\mathbb{E}$ and , then $\sqrt{r}\in\mathbb{E}$ .

The field $\mathbb{E}$ can be extended in a natural manner to a subfield of $\mathbb{C}$ that is not a subfield of $\mathbb{R}$ . Let $\mathbb{F}$ be a subset of $\mathbb{C}$ that has the following properties:

$0,1\in\mathbb{F}$ ;
If $a,b\in\mathbb{F}$ , then also $a\pm b$ , , and $a/b\in\mathbb{F}$ , the last of which is meaningful only when $b\not=0$ ;
If $z\in\mathbb{F} \setminus \{0\}$ and $\operatorname{arg}(z)=\theta$ where $0 \le \theta < 2\pi$ , then $\sqrt{\vert z\vert}e^{\frac{i\theta}{2}}\in\mathbb{F}$ .

Then $\mathbb{F}$ is the field of constructible numbers.

Note that $\mathbb{E}\subset\mathbb{F}$ . Moreover, $\mathbb{F}\cap\mathbb{R}=\mathbb{E}$ .

An element of $\mathbb{F}$ is called a constructible number. These numbers can be “constructed” by a process that will be described shortly.

Conversely, let us start with a subset of $\mathbb{C}$ such that contains a non-zero complex number. Call any of the binary operations in condition 2 as well as the square root unary operation in condition 3 a ruler and compass operation. Call a complex number constructible from if it can be obtained from elements of by a finite sequence of ruler and compass operations. Note that $1\in S$ . If $S^{\prime}$ is the set of numbers constructible from using only the binary ruler and compass operations (those in condition 2), then $S^{\prime}$ is a subfield of $\mathbb{C}$ , and is the smallest field containing . Next, denote $\hat{S}$ the set of all constructible numbers from . It is not hard to see that $\hat{S}$ is also a subfield of $\mathbb{C}$ , but an extension of $S^{\prime}$ . Furthermore, it is not hard to show that $\hat{S}$ is Euclidean. The general process (algorithm) of elements in $\hat{S}$ from elements in using finite sequences of ruler and compass operations is called a 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 , 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 $\hat{S}=\mathbb{F}$ is the field of constructible numbers.

Note that the lengths of constructible line segments on the Euclidean plane are exactly the positive elements of $\mathbb{E}$ . Note also that the set $\mathbb{F}$ is in one-to-one correspondence with the set of constructible points on the Euclidean plane. These facts provide a connection between abstract algebra and compass and straightedge constructions.

"constructible numbers" is owned by CWoo. [ full author list (2) | owner history (1) ]

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Also defines:

ruler and compass operation, compass and ruler operation, compass and straightedge operation, straightedge and compass operation, constructible number, constructible from, constructible, field of constructible numbers, field of real constructible numbers

Attachments:: motivation of definition of constructible numbers (Topic) by Wkbj79

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Cross-references: algebra, one-to-one correspondence, positive elements, Euclidean plane, lengths, rational number, compass, ruler, real number, points, ruler and compass construction, algorithm, extension, binary, finite sequence, operation, unary, square root, binary operations, complex number, contains, numbers, subset, field, properties, Euclidean, subfield
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This is version 14 of constructible numbers, born on 2007-06-13, modified 2007-06-25.
Object id is 9583, canonical name is ConstructibleNumbers.
Accessed 2659 times total.

Classification:

AMS MSC:

12D15 (Field theory and polynomials :: Real and complex fields :: Fields related with sums of squares )

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