units of real cubic fields with exactly one real embedding

units of real cubic fields with exactly one real embedding

Let

K

⊆

ℝ

be a

number field

with

[

K

:

ℚ

]

=

3

such that

K

has exactly one

real embedding

. Thus,

r

=

1

and

s

=

1

. Let

𝒪

K

*

denote the

group of units

of the

ring of integers

of

K

. By Dirichlet’s unit theorem,

𝒪

K

*

≅

μ

⁢

(

K

)

×

ℤ

since

r

+

s

-

1

=

1

. The only

roots of unity

in

K

are

1

and

-

1

because

K

⊆

ℝ

. Thus,

μ

⁢

(

K

)

=

{

1

,

-

1

}

. Therefore, there exists

u

∈

𝒪

K

*

with

u

>

1

, such that every element of

𝒪

K

*

is of the form

±

u

n

for some

n

∈

ℤ

.

Let

ρ

>

0

and

0

<

θ

<

π

such that the

conjugates

of

u

are

ρ

⁢

e

i

⁢

θ

and

ρ

⁢

e

-

i

⁢

θ

. Since

u

is a unit,

N

⁢

(

u

)

=

±

1

. Thus,

±

1

=

N

⁢

(

u

)

=

u

⁢

(

ρ

⁢

e

i

⁢

θ

)

⁢

(

ρ

⁢

e

-

i

⁢

θ

)

=

u

⁢

ρ

2

. Since

u

>

0

and

ρ

2

>

0

, it must be the case that

u

⁢

ρ

2

=

1

. Thus,

u

=

1

ρ

2

. One can then deduce that

disc

⁡

u

=

-

4

⁢

sin

2

⁡

θ

⁢

(

ρ

3

+

1

ρ

3

-

2

⁢

cos

⁡

θ

)

2

. Since the maximum value of the

polynomial

4

⁢

sin

2

⁡

θ

⁢

(

x

-

2

⁢

cos

⁡

θ

)

2

-

4

⁢

x

2

is at most

16

, one can deduce that

|

disc

⁡

u

|

≤

4

⁢

(

u

3

+

1

u

3

+

4

)

. Define

d

=

|

disc

⁡

𝒪

K

|

. Then

d

≤

|

disc

⁡

u

|

≤

4

⁢

(

u

3

+

1

u

3

+

4

)

. Thus,

u

3

≥

d

4

-

4

-

1

u

3

. From this, one can obtain that

u

3

≥

d

-

16

+

d

2

-

32

⁢

d

+

192

8

. (Note that a higher

lower bound

on

u

3

is desirable, and the one stated here is much higher than that stated in Marcus.) Thus,

u

2

≥

(

d

-

16

+

d

2

-

32

⁢

d

+

192

8

)

2

3

. Therefore, if an element

x

∈

𝒪

K

*

can be found such that

1

<

x

<

(

d

-

16

+

d

2

-

32

⁢

d

+

192

8

)

2

3

, then

x

=

u

.

Following are some applications:

•

The above is most applicable for finding the

fundamental unit

of a ring of integers of a

pure cubic field

. For example, if

K

=

ℚ

⁢

(

2

3

)

, then

d

=

108

, and the lower bound on

u

2

is

(

23

+

10

⁢

21

2

)

2

3

, which is larger than

9

. Note that

(

4

3

+

2

3

+

1

)

⁢

(

2

3

-

1

)

=

2

-

1

=

1

. Since

1

<

4

3

+

2

3

+

1

<

9

, it follows that

4

3

+

2

3

+

1

is the fundamental unit of

𝒪

K

.

•

The above can also be used for any number field

K

with

[

K

:

ℚ

]

=

3

such that

K

has exactly one real embedding. Let

σ

be the real embedding. Then the above produces the fundamental unit

u

of

σ

⁢

(

K

)

. Thus,

σ

-

1

⁢

(

u

)

is a fundamental unit of

K

.

References

1

Marcus, Daniel A.

Number Fields

. New York: Springer-Verlag, 1977.

Title

units of real cubic fields with exactly one real embedding

Canonical name

UnitsOfRealCubicFieldsWithExactlyOneRealEmbedding

Date of creation

2013-03-22 16:02:25

Last modified on

2013-03-22 16:02:25

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Application

Classification

msc 11R27

Classification

msc 11R16

Classification

msc 11R04

Related topic

NormAndTraceOfAlgebraicNumber

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