index of the group of cyclotomic units in the full unit group

index of the group of cyclotomic units in the full unit group

Let

K

n

=

ℚ

⁢

(

ζ

p

n

)

where

ζ

p

n

is a

primitive

p

n

th

root of unity

, let

h

n

be the

class number

of

K

n

and
let

𝒪

n

=

𝒪

K

n

be the

ring of integers

in

K

n

. Let

E

n

=

𝒪

n

×

be the

group of units

in

K

n

. The

cyclotomic units

are a

subgroup

C

n

of

E

n

which
satisfy:

•

The elements of

C

n

are defined analytically.

•

The subgroup

C

n

is of finite index in

E

n

. Furthermore,
the index is

h

n

+

: Let

E

n

+

be the group of units in

K

n

+

and let

C

n

+

=

C

n

∩

E

n

+

. Then

[

E

n

+

:

C

n

+

]

=

h

n

+

. Moreover, it can be shown that

[

E

n

:

C

n

]

=

[

E

n

+

:

C

n

+

]

because

E

n

=

μ

p

n

⁢

E

n

+

(this is
exercise 8.5 in

[

1

]

).

•

The subgroups

C

n

behave “well” in towers. More
precisely, the norm of

C

n

+

1

down to

K

n

is

C

n

. This
follows from the fact that the norm of

ζ

p

n

+

1

down to

K

n

is

ζ

p

n

.

Definition 1

.

Let

p

be prime and let

n

≥

1

. Let

ζ

p

n

be a
primitive

p

n

th root of unity.

1.

The cyclotomic unit group

C

n

+

⊂

K

n

+

=

ℚ

⁢

(

ζ

p

n

)

+

is the group of units generated by

-

1

and the units

ξ

a

=

ζ

p

n

(

1

-

a

)

/

2

⁢

1

-

ζ

p

n

a

1

-

ζ

p

n

=

±

sin

⁡

(

π

⁢

a

/

p

n

)

sin

⁡

(

π

/

p

n

)

with

1

<

a

<

p

n

2

and

gcd

⁡

(

a

,

p

)

=

1

.

2.

The cyclotomic unit group

C

n

⊂

K

n

=

ℚ

⁢

(

ζ

p

n

)

is the group generated by

ζ

p

n

and the cyclotomic units

C

n

+

of

K

n

+

.

Remark 1

.

Let

σ

a

:

ζ

p

n

→

ζ

p

n

a

be an element of

Gal

⁡

(

K

n

/

ℚ

)

. Then:

ξ

a

=

ζ

p

n

(

1

-

a

)

/

2

⁢

1

-

ζ

p

n

a

1

-

ζ

p

n

=

(

ζ

p

n

-

1

/

2

⁢

(

1

-

ζ

p

n

)

)

σ

a

ζ

p

n

-

1

/

2

⁢

(

1

-

ζ

p

n

)

.

Remark 2

.

Let

g

be a

primitive root

modulo

p

n

. Let

a

≡

g

r

mod

p

n

. Then one can rewrite

ξ

a

as:

ξ

a

=

∏

i

=

0

r

-

1

ξ

g

σ

g

i

.

In particular

ξ

g

generates

C

n

+

/

{

±

1

}

as a module over

ℤ

⁢

[

Gal

⁡

(

ℚ

⁢

(

ζ

p

n

)

+

/

ℚ

)

]

.

Notice that in order to show that the index of

C

n

in

K

n

is
finite it suffices to show that the index of

C

n

+

in

K

n

+

is
finite. Indeed, let

[

K

n

:

ℚ

]

=

2

d

. Since

K

n

is a

totally
imaginary field

and by Dirichlet’s unit theorem the free rank of

E

n

is

r

1

+

r

2

-

1

=

d

-

1

. On the other hand,

[

K

n

+

:

ℚ

]

=

d

and

K

n

+

is totally real, thus the free rank of

E

n

+

is also

d

-

1

. Therefore the free rank of

E

n

+

and

E

n

are equal. As
we claimed before, the index

[

E

n

+

:

C

n

+

]

is rather interesting
to us.

Theorem 1

(

[

1

]

,Thm. 8.2).

Let

p

be a prime and

n

≥

1

. Let

h

n

+

be the class number of

Q

⁢

(

ζ

p

n

)

+

. The cyclotomic
units

C

n

+

of

Q

⁢

(

ζ

p

n

)

+

are a subgroup of finite
index in the full unit group

E

n

+

. Furthermore:

h

n

+

=

[

E

n

+

:

C

n

+

]

=

[

E

n

:

C

n

]

.

In the proof of the previous theorem one calculates the

regulator

of the units

ξ

a

in terms of values of

Dirichlet L-functions

with even

characters

. In particular, one calculates:

R

⁢

(

{

ξ

a

}

)

=

±

∏

χ

≠

χ

0

1

2

⁢

τ

⁢

(

χ

)

⁢

L

⁢

(

1

,

χ

¯

)

=

h

n

+

⋅

R

+

where in the last equality one uses the properties of

Gauss
sums

and the

class number formula

in terms of Dirichlet
L-functions evaluated at

s

=

1

. This yields that

R

⁢

(

{

ξ

a

}

)

in
non-zero, therefore the index in

E

n

+

is finite and moreover

h

n

+

=

R

⁢

(

{

ξ

a

}

)

R

+

=

[

E

n

+

:

C

n

+

]

=

[

E

n

:

C

n

]

.

An immediate consequence of this is that if

p

divides

h

n

+

then there exists a cyclotomic unit

γ

∈

C

n

+

such that

γ

is a

p

th power in

E

n

+

but not in

C

n

+

.

References

1

L. C. Washington,

Introduction to

Cyclotomic
Fields

, Second Edition, Springer-Verlag, New York.

Title

index of the group of cyclotomic units in the full unit group

Canonical name

IndexOfTheGroupOfCyclotomicUnitsInTheFullUnitGroup

Date of creation

2013-03-22 15:42:49

Last modified on

2013-03-22 15:42:49

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11R18

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