Archimedean ordered fields are real

Archimedean ordered fields are real

In this entry, we shall show that every

Archimedean

ordered field

is

isomorphic

to
a

subfield

of the

field of real

numbers. To accomplish this, we shall

construct

an

isomorphism

using

Dedekind cuts

.

As a preliminary, we agree to some conventions. Let

𝔽

denote an ordered
field with

ordering relation

>

. We will identify the integers with

multiples

of
the

multiplicative identity

of fields. We further assume that

𝔽

satisfies

the

Archimedean property

: for every

element

x

of

𝔽

, there exists an
integer

n

such that

x

<

n

.

Since

𝔽

is an ordered field, it must have characteristic zero. Hence,
the subfield generated by the multiplicative identity is isomorphic to the

field
of rational

numbers. Following the convention proposed above, we will identify
this subfield with

ℚ

. We use this subfield to construct a map

ρ

from

𝔽

to

ℝ

:

Definition 1

.

For every

x

∈

F

, we define

ρ

⁢

(

x

)

=

(

{

y

∈

ℚ

∣

y

>

x

}

,

{

y

∈

ℚ

∣

y

≤

x

}

)

Theorem 1

.

For every

x

∈

F

, we find that

ρ

⁢

(

x

)

is a Dedekind cut.

Proof.

Because, for all

y

∈

ℚ

, we have either

y

>

x

or

y

≤

x

, the two
sets of

ρ

⁢

(

x

)

form a

partition

of

ℚ

. Furthermore, every element
of the latter set is less than every element of the former set. By the Archimedean
property, there exists an integer

n

such that

x

<

n

; hence the former set is
not empty. Likewise, there exists and integer

m

such that

-

x

<

m

, or

x

>

-

m

,
so the latter set is also not empty.
∎

Having seen that

ρ

is a bona fide map into the

real numbers

, we now show that
it is not just any old map, but a

monomorphism

of fields.

Theorem 2

.

The map

ρ

:

F

→

R

is a monomorphism.

Proof.

Let

p

and

q

be elements of

𝔽

; set

(

A

,

B

)

=

ρ

⁢

(

p

)

, set

(

C

,

D

)

=

ρ

⁢

(

q

)

, and set

(

E

,

F

)

=

ρ

⁢

(

p

+

q

)

. Since

a

>

p

and

b

>

q

implies

a

+

b

>

p

+

q

for all

rational numbers

a

and

b

, it follows that

a

∈

A

and

b

∈

C

implies that

a

+

b

∈

E

. Likewise, since

a

≤

p

and

b

≤

q

implies

a

+

b

≤

p

+

q

for all rational numbers

a

and

b

, it follows that

a

∈

B

and

b

∈

D

implies

a

+

b

∈

F

. Hence,

ρ

⁢

(

p

)

+

ρ

⁢

(

q

)

=

ρ

⁢

(

p

+

q

)

.

Since a rational number is

positive

if and only if it is greater than

0

, it
follows that

ρ

⁢

(

0

)

=

0

. Together with the fact proven in the last paragraph,
this implies that

ρ

⁢

(

-

x

)

=

-

ρ

⁢

(

x

)

for all

x

∈

𝔽

.

Suppose that

p

and

q

are

positive elements

of

𝔽

. As before, set

(

A

,

B

)

=

ρ

⁢

(

p

)

, set

(

C

,

D

)

=

ρ

⁢

(

q

)

, and set

(

E

,

F

)

=

ρ

⁢

(

p

⋅

q

)

. Since

a

>

p

and

b

>

q

implies

a

⋅

b

>

p

⋅

q

for all rational numbers

a

and

b

, it follows that

a

∈

A

and

b

∈

C

implies that

a

⋅

b

∈

E

.
Likewise, since

a

≤

p

and

b

≤

q

implies

a

⋅

b

≤

p

⋅

q

for all rational numbers

a

and

b

, it follows that

a

∈

B

and

b

∈

D

implies

a

⋅

b

∈

F

. Hence,

ρ

⁢

(

p

)

⋅

ρ

⁢

(

q

)

=

ρ

⁢

(

p

⋅

q

)

.

By using the fact demonstrated previously that

ρ

⁢

(

-

x

)

=

-

ρ

⁢

(

x

)

, we may extend
what was shown above to the statement that

ρ

⁢

(

p

⋅

q

)

=

ρ

⁢

(

p

)

⋅

ρ

⁢

(

q

)

for all

p

,

q

∈

𝔽

. Thus,

ρ

is a

morphism

of fields. Since

𝔽

is Archmiedean, if

p

≠

q

, there must exist a rational number

r

between

p

and

q

,
hence

ρ

⁢

(

p

)

≠

ρ

⁢

(

q

)

, so

ρ

is a monomorphism.
∎

Since

ρ

is a morphism of fields, its image is a

subring

of

ℝ

. Since

ρ

is a monomorphism, its

restriction

to this image is an isomorphism, hence

𝔽

is
isomorphic to a subfield of

ℝ

.

Title

Archimedean ordered fields are real

Canonical name

ArchimedeanOrderedFieldsAreReal

Date of creation

2013-03-22 17:26:22

Last modified on

2013-03-22 17:26:22

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 12D99

Related topic

GelfandTornheimTheorem

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