Chapter 7
The Structure of
Z
*
n
We study the structure of the group of units
Z
*
n
of the ring
Z
n
. As we know,
Z
*
n
consists of those
elements [
a
mod
n
]
∈
Z
n
such that
a
is an integer relatively prime to
n
.
Suppose
n
=
p
e
1
1
· · ·
p
e
r
r
is the factorization of
n
into primes. By the Chinese Remainder Theo
rem, we have the ring isomorphism
Z
n
∼
=
Z
p
e
1
1
× · · · ×
Z
p
er
r
which induces a group isomorphism
Z
*
n
∼
=
Z
*
p
e
1
1
× · · · ×
Z
*
p
er
r
.
Thus the problem of studying the group of units of modulo an arbitrary integer reduces to the
studying the group of units modulo a prime power.
Define
φ
(
n
) to be the cardinality of
Z
*
n
. This is equal to the number of integers in the interval
{
0
, . . . , n

1
}
that are relatively prime to
n
. It is clear that
φ
(
p
) =
p

1 for prime
p
.
Theorem 7.1
If
n
=
p
e
1
1
· · ·
p
e
r
r
is the prime factorization of
n
, then
φ
(
n
) =
p
e
1

1
1
(
p
1

1)
· · ·
p
e
r

1
r
(
p
r

1)
.
Proof.
By the Chinese Remainder Theorem, we have
φ
(
n
) =
φ
(
p
e
1
1
)
· · ·
φ
(
p
e
r
r
), so it suffices to
show that for a prime power
p
e
,
φ
(
p
e
) =
p
e

1
(
p

1). Now,
φ
(
p
e
) is equal to
p
e
minus the number
of integers in the interval
{
0
, . . . , p
e

1
}
that are a multiple of
p
.
The integers in the interval
{
0
, . . . , p
e

1
}
that are multiplies of
p
are precisely 0
, p,
2
p,
3
p, . . . ,
(
p
e

1

1)
p
, of which there are
p
e

1
. Thus,
φ
(
p
e
) =
p
e

p
e

1
=
p
e

1
(
p

1).
2
Next, we study the structure of the group
Z
*
n
. Again, by the Chinese Remainder Theorem, it
suffices to consider
Z
*
p
e
for prime
p
.
We consider first consider the simpler case
Z
*
p
.
Theorem 7.2
Z
*
p
is a cyclic group.
This theorem follows from the more general theorem:
Theorem 7.3
Let
K
be a field and
G
a subgroup of
K
*
of finite order. Then
G
is cyclic.
48