constructible numbers

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:

1.

$0,1\in\mathbb{E}$ ;
2.

If $a,b\in\mathbb{E}$ , then also $a\pm b$ , $a b$ , and $a/b\in\mathbb{E}$ , the last of which is meaningful only when $b\not=0$ ;
3.

If $r\in\mathbb{E}$ and $r>0$ , 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:

1.

$0,1\in\mathbb{F}$ ;
2.

If $a,b\in\mathbb{F}$ , then also $a\pm b$ , $a b$ , and $a/b\in\mathbb{F}$ , the last of which is meaningful only when $b\not=0$ ;
3.

If $z\in\mathbb{F}\setminus\{0\}$ and $\operatorname{arg}(z)=\theta$ where $0\leq\theta<2\pi$ , then $\sqrt{|z|}e^{\frac{i\theta}{2}}\in\mathbb{F}$ .

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 $S$ of $\mathbb{C}$ such that $S$ 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 $S$ 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 $\mathbb{C}$ , 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 $\mathbb{C}$ , 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 $S$ 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 $S$ , one can use a ruler and compass to construct these elements of $\hat{S}$ .

If $S=\{1\}$ (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 (http://planetmath.org/Constructible2) 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 (http://planetmath.org/Constructible2) on the Euclidean plane. These facts provide a between abstract algebra and compass and straightedge constructions.

Title	constructible numbers
Canonical name	ConstructibleNumbers
Date of creation	2013-03-22 17:15:01
Last modified on	2013-03-22 17:15:01
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	17
Author	CWoo (3771)
Entry type	Definition
Classification	msc 12D15
Related topic	EuclideanField
Related topic	CompassAndStraightedgeConstruction
Related topic	TheoremOnConstructibleAngles
Related topic	TheoremOnConstructibleNumbers
Defines	ruler and compass operation
Defines	compass and ruler operation
Defines	compass and straightedge operation
Defines	straightedge and compass operation
Defines	constructible number
Defines	constructible from
Defines	constructible
Defines	field of constructible numbers
Defines	field of real constructible numbers