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where
ζ
is a complex primitive
d
th root of 1. An element in
R
is a unit if
and only if its norm is 1 or
−
1. Moreover, the norm function is multiplicative:
N
(
ab
)=
N
(
a
)
N
(
b
) for all
a, b
∈
R
.
Lemma 1.
We have
1.
∀
z
∈
R
:
|
R/
(
z
)
|
=
|
N
(
z
)
|
.
2.
∀
z
∈
R
:log
p
|
N
(
z
)
|≤
d
log
p
(
d
)+
λ
(
z
)
.
3. If
d
=2
,then
λ
(
z
)
.
4. If
d
=2
and
R
is the full ring of integers in its fraction field, then there
is a constant
c
p
∈
R
∀
z
∈
R
:log
p
|
N
(
z
)
|≤
+
so that
∀
z
∈
R
there is a unit
u
∈
R
:
λ
(
uz
)
≤
2log
p
(
|
N
(
z
)
|
+
c
p
)
−
log
p
(4)
.
5. If
d
=
p
=2
,then
∀
z
∈
R
there is a unit
u
∈
R
so that
λ
(
uz
)
≤
log
2
|
N
(
z
)
|
+1
.
It follows that if
d
=2and
λ
(
z
) is minimal among all associates of
z
,then
log
p
|
+
e
p
for some
constant
e
p
.Moreover,if
p
=2and
λ
(
z
) is minimal among all associates of
z
,
then
λ
(
z
)
N
(
z
)
|≤
λ
(
z
)
≤
2log
p
(
|
N
(
z
)
|
+
c
p
)
−
log
p
(4)
≤
2log
p
|
N
(
z
)
|
λ
(
z
)+1.
The ring
R
is Noetherian. Thus every element
q
≤
log
2
|
N
(
z
)
|
+1
≤
∈
R
can be written as a
product
z
=
u
z
e
i
,where
u
is a unit, the
z
i
are irreducible, each
e
i
is positive,
and
z
i
and
z
j
are not associates
3
if
i
=
j
(this representation may not be unique).
We may assume that
λ
(
z
i
) is minimal among all associates of
z
i
.Inparticular
we can write
t
z
e
i
i
π
n
−
1=
u
,
(3)
i
=1
where
u
is a unit and each
z
i
is irreducible.
Suppose further that
R
is a unique factorization domain or UFD. The rep-
resentation of
q
as a product of irreducibles is then unique up to permutation
of the
z
i
and replacing a
z
i
by an associate (and so also changing the unit
u
).
In this case there is a connection element for any sequence that is minimal in
the sense that it divides all other connection elements for the sequence. Any two
minimal connection elements are associates.
Lemma 2.
R
is not a unit. Let
u/z
and
v/z
be rational elements whose
π
-adic expansions are periodic. If
z
is the
minimal connection element for either of these sequences, then
u
and
v
are not
congruent modulo
z
.
Suppose
R
=
Z
[
π
]
is a UFD and
z
∈
It follows that the set
U
z
of elements
u
R
such that
u/z
has a strictly periodic
π
-adic expansion and such that
z
is the minimal connection element is a subset
of a complete set of representatives
V
z
modulo
z
.Let
u ∈ R
.Then
u/z
has
an eventually periodic
π
-adic expansion, so there is an element
a ∈ R
so that
y/z
=
a
+
u/z
has a strictly periodic
π
-adic expansion. Then
y
=
u
+
az
,sowe
may assume that if
u
∈
∈
V
z
,then
u/z
has a strictly periodic
π
-adic expansion.
3
Elements
a
and
b
are
associates
if
a
=
vb
for some unit
v
.
