constructible numbers

constructible numbers

The smallest

subfield

𝔼

of

ℝ

over

ℚ

such that

𝔼

is

Euclidean

is called the

field of real constructible numbers

. First, note that

𝔼

has the following properties:

1.

0

,

1

∈

𝔼

;

2.

If

a

,

b

∈

𝔼

, then also

a

±

b

,

a

⁢

b

, and

a

/

b

∈

𝔼

, the last of which is meaningful only when

b

≠

0

;

3.

If

r

∈

𝔼

and

r

>

0

, then

r

∈

𝔼

.

The field

𝔼

can be extended in a natural manner to a subfield of

ℂ

that is not a subfield of

ℝ

. Let

𝔽

be a subset of

ℂ

that has the following properties:

1.

0

,

1

∈

𝔽

;

2.

If

a

,

b

∈

𝔽

, then also

a

±

b

,

a

⁢

b

, and

a

/

b

∈

𝔽

, the last of which is meaningful only when

b

≠

0

;

3.

If

z

∈

𝔽

∖

{

0

}

and

arg

⁡

(

z

)

=

θ

where

0

≤

θ

<

2

⁢

π

, then

|

z

|

⁢

e

i

⁢

θ

2

∈

𝔽

.

Then

𝔽

is the

field of constructible numbers

.

Note that

𝔼

⊂

𝔽

. Moreover,

𝔽

∩

ℝ

=

𝔼

.

An element of

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

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

∈

S

. If

S

′

is the set of numbers constructible from

S

using only the binary ruler and compass operations (those in condition 2), then

S

′

is a subfield of

ℂ

, and is the smallest field containing

S

. Next, denote

S

^

the set of all constructible numbers from

S

. It is not hard to see that

S

^

is also a subfield of

ℂ

, but an

extension

of

S

′

. Furthermore, it is not hard to show that

S

^

is Euclidean. The general process (algorithm) of

elements

in

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

S

^

.

If

S

=

{

1

}

(or any

rational number

), we see that

S

^

=

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

𝔼

. Note also that the set

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

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