constructible numbers
The smallest subfield^{} $\mathrm{\pi \x9d\x94\u038c}$ of $\mathrm{\beta \x84\x9d}$ over $\mathrm{\beta \x84\x9a}$ such that $\mathrm{\pi \x9d\x94\u038c}$ is Euclidean^{} is called the field of real constructible numbers. First, note that $\mathrm{\pi \x9d\x94\u038c}$ has the following properties:

1.
$0,1\beta \x88\x88\mathrm{\pi \x9d\x94\u038c}$;

2.
If $a,b\beta \x88\x88\mathrm{\pi \x9d\x94\u038c}$, then also $a{\rm B}\pm b$, $a\beta \x81\u2019b$, and $a/b\beta \x88\x88\mathrm{\pi \x9d\x94\u038c}$, the last of which is meaningful only when $b\beta \x890$;

3.
If $r\beta \x88\x88\mathrm{\pi \x9d\x94\u038c}$ and $r>0$, then $\sqrt{r}\beta \x88\x88\mathrm{\pi \x9d\x94\u038c}$.
The field $\mathrm{\pi \x9d\x94\u038c}$ can be extended in a natural manner to a subfield of $\mathrm{\beta \x84\x82}$ that is not a subfield of $\mathrm{\beta \x84\x9d}$. Let $\mathrm{\pi \x9d\x94\xbd}$ be a subset of $\mathrm{\beta \x84\x82}$ that has the following properties:

1.
$0,1\beta \x88\x88\mathrm{\pi \x9d\x94\xbd}$;

2.
If $a,b\beta \x88\x88\mathrm{\pi \x9d\x94\xbd}$, then also $a{\rm B}\pm b$, $a\beta \x81\u2019b$, and $a/b\beta \x88\x88\mathrm{\pi \x9d\x94\xbd}$, the last of which is meaningful only when $b\beta \x890$;

3.
If $z\beta \x88\x88\mathrm{\pi \x9d\x94\xbd}\beta \x88\x96\{0\}$ and $\mathrm{arg}\beta \x81\u2018(z)=\mathrm{\Xi \u0388}$ where $$, then $\sqrt{z}\beta \x81\u2019{e}^{\frac{i\beta \x81\u2019\mathrm{\Xi \u0388}}{2}}\beta \x88\x88\mathrm{\pi \x9d\x94\xbd}$.
Then $\mathrm{\pi \x9d\x94\xbd}$ is the field of constructible numbers.
Note that $\mathrm{\pi \x9d\x94\u038c}\beta \x8a\x82\mathrm{\pi \x9d\x94\xbd}$. Moreover, $\mathrm{\pi \x9d\x94\xbd}\beta \x88\copyright \mathrm{\beta \x84\x9d}=\mathrm{\pi \x9d\x94\u038c}$.
An element of $\mathrm{\pi \x9d\x94\xbd}$ is called a constructible number. These numbers can be βconstructedβ by a process that will be described shortly.
Conversely, let us start with a subset $S$ of $\mathrm{\beta \x84\x82}$ such that $S$ contains a nonzero 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 $1\beta \x88\x88S$. If ${S}^{\beta \x80\xb2}$ is the set of numbers constructible from $S$ using only the binary ruler and compass operations (those in condition 2), then ${S}^{\beta \x80\xb2}$ is a subfield of $\mathrm{\beta \x84\x82}$, and is the smallest field containing $S$. Next, denote $\widehat{S}$ the set of all constructible numbers from $S$. It is not hard to see that $\widehat{S}$ is also a subfield of $\mathrm{\beta \x84\x82}$, but an extension^{} of ${S}^{\beta \x80\xb2}$. Furthermore, it is not hard to show that $\widehat{S}$ is Euclidean. The general process (algorithm) of elements^{} in $\widehat{S}$ 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 nonzero real number in $S$, one can use a ruler and compass to construct these elements of $\widehat{S}$.
If $S=\{1\}$ (or any rational number^{}), we see that $\widehat{S}=\mathrm{\pi \x9d\x94\xbd}$ 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 $\mathrm{\pi \x9d\x94\u038c}$. Note also that the set $\mathrm{\pi \x9d\x94\xbd}$ is in onetoone 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  20130322 17:15:01 
Last modified on  20130322 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 