2Rings
IB Groups, Rings and Modules
2.6 Gaussian integers
We’ve mentioned the Gaussian integers already.
Definition (Gaussian integers). The Gaussian integers is the subring
Z[i] = {a + bi : a, b ∈ Z} ≤ C.
We have already shown that the norm
N
(
a
+
ib
) =
a
2
+
b
2
is a Euclidean
function for
Z
[
i
]. So
Z
[
i
] is a Euclidean domain, hence principal ideal domain,
hence a unique factorization domain.
Since the units must have norm 1, they are precisely
±
1
, ±i
. What does
factorization in Z[i] look like? What are the primes? We know we are going to
get new primes, i.e. primes that are not integers, while we will lose some other
primes. For example, we have
2 = (1 + i)(1 − i).
So 2 is not irreducible, hence not prime. However, 3 is a prime. We have
N
(3) = 9. So if 3 =
uv
, with
u, v
not units, then 9 =
N
(
u
)
N
(
v
), and neither
N
(
u
) nor
N
(
v
) are 1. So
N
(
u
) =
N
(
v
) = 3. However, 3 =
a
2
+
b
2
has no
solutions with
a, b ∈ Z
. So there is nothing of norm 3. So 3 is irreducible, hence
a prime.
Also, 5 is not prime, since
5 = (1 + 2i)(1 − 2i).
How can we understand which primes stay as primes in the Gaussian integers?
Proposition. A prime number
p ∈ Z
is prime in
Z
[
i
] if and only if
p 6
=
a
2
+
b
2
for a, b ∈ Z \{0}.
The proof is exactly what we have done so far.
Proof. If p = a
2
+ b
2
, then p = (a + ib)(a − ib). So p is not irreducible.
Now suppose
p
=
uv
, with
u, v
not units. Taking norms, we get
p
2
=
N
(
u
)
N
(
v
). So if
u
and
v
are not units, then
N
(
u
) =
N
(
v
) =
p
. Writing
u = a + ib, then this says a
2
+ b
2
= p.
So what we have to do is to understand when a prime
p
can be written as a
sum of two squares. We will need the following helpful lemma:
Lemma. Let
p
be a prime number. Let
F
p
=
Z/pZ
be the field with
p
elements.
Let
F
×
p
=
F
p
\{
0
}
be the group of invertible elements under multiplication. Then
F
×
p
∼
=
C
p−1
.
Proof.
Certainly
F
×
p
has order
p −
1, and is abelian. We know from the classifi-
cation of finite abelian groups that if
F
×
p
is not cyclic, then it must contain a
subgroup
C
m
×C
m
for
m >
1 (we can write it as
C
d
×C
d
0
×···
, and that
d
0
| d
.
So C
d
has a subgroup isomorphic to C
d
0
).
We consider the polynomial
X
m
−
1
∈ F
p
[
x
], which is a UFD. At best, this
factors into
m
linear factors. So
X
m
−
1 has at most
m
distinct roots. But if
C
m
× C
m
≤ F
×
p
, then we can find
m
2
elements of order dividing
m
. So there
are
m
2
elements of
F
p
which are roots of
X
m
−
1. This is a contradiction. So
F
×
p
is cyclic.
This is a funny proof, since we have not found any element that has order
p − 1.
Proposition. The primes in Z[i] are, up to associates,
(i) Prime numbers p ∈ Z ≤ Z[i] such that p ≡ 3 (mod 4).
(ii)
Gaussian integers
z ∈ Z
[
i
] with
N
(
z
) =
z¯z
=
p
for some prime
p
such that
p = 2 or p ≡ 1 (mod 4).
Proof.
We first show these are primes. If
p ≡
3 (
mod
4), then
p 6
=
a
2
+
b
2
, since
a square number mod 4 is always 0 or 1. So these are primes in Z[i].
On the other hand, if
N
(
z
) =
p
, and
z
=
uv
, then
N
(
u
)
N
(
v
) =
p
. So
N
(
u
)
is 1 or
N
(
v
) is 1. So
u
or
v
is a unit. Note that we did not use the condition
that p 6≡ 3 (mod 4). This is not needed, since N(z) is always a sum of squares,
and hence N (z) cannot be a prime that is 3 mod 4.
Now let
z ∈ Z
[
i
] be irreducible, hence prime. Then
¯z
is also irreducible. So
N
(
z
) =
z¯z
is a factorization of
N
(
z
) into irreducibles. Let
p ∈ Z
be an ordinary
prime number dividing N (z), which exists since N(z) 6= 1.
Now if
p ≡
3 (
mod
4), then
p
itself is prime in
Z
[
i
] by the first part of the
proof. So
p | N
(
z
) =
z¯z
. So
p | z
or
p | ¯z
. Note that if
p | ¯z
, then
p | z
by taking
complex conjugates. So we get
p | z
. Since both
p
and
z
are irreducible, they
must be equal up to associates.
Otherwise, we get
p
= 2 or
p ≡
1 (
mod
4). If
p ≡
1 (
mod
4), then
p −
1 = 4
k
for some
k ∈ Z
. As
F
×
p
∼
=
C
p−1
=
C
4k
, there is a unique element of order 2 (this
is true for any cyclic group of order 4
k
— think
Z/
4
kZ
). This must be [
−
1]
∈ F
p
.
Now let a ∈ F
×
p
be an element of order 4. Then a
2
has order 2. So [a
2
] = [−1].
This is a complicated way of saying we can find an
a
such that
p | a
2
+ 1.
Thus
p |
(
a
+
i
)(
a − i
). In the case where
p
= 2, we know by checking directly
that 2 = (1 + i)(1 − i).
In either case, we deduce that
p
(or 2) is not prime (hence irreducible),
since it clearly does not divide
a ± i
(or 1
± i
). So we can write
p
=
z
1
z
2
, for
z
1
, z
2
∈ Z[i] not units. Now we get
p
2
= N(p) = N(z
1
)N(z
2
).
As the
z
i
are not units, we know
N
(
z
1
) =
N
(
z
2
) =
p
. By definition, this means
p = z
1
¯z
1
= z
2
¯z
2
. But also p = z
1
z
2
. So we must have ¯z
1
= z
2
.
Finally, we have
p
=
z
1
¯z
1
| N
(
z
) =
z¯z
. All these
z
,
z
i
are irreducible. So
z
must be an associate of z
1
(or maybe ¯z
1
). So in particular N(z) = p.
Corollary. An integer
n ∈ Z
≥0
may be written as
x
2
+
y
2
(as the sum of two
squares) if and only if “when we write
n
=
p
n
1
1
p
n
2
2
···p
n
k
k
as a product as distinct
primes, then p
i
≡ 3 (mod 4) implies n
i
is even”.
We have proved this in the case when n is a prime.
Proof. If n = x
2
+ y
2
, then we have
n = (x + iy)(x − iy) = N (x + iy).
Let
z
=
x
+
iy
. So we can write
z
=
α
1
···α
q
as a product of irreducibles in
Z
[
i
].
By the proposition, each
α
i
is either
α
i
=
p
(a genuine prime number with
p ≡
3
(
mod
4)), or
N
(
α
i
) =
p
is a prime number which is either 2 or
≡
1 (
mod
4). We
now take the norm to obtain
N = x
2
+ y
2
= N(z) = N(α
1
)N(α
2
) ···N(α
q
).
Now each
N
(
α
i
) is either
p
2
with
p ≡
3 (
mod
4), or is just
p
for
p
= 2 or
p ≡
1
(
mod
4). So if
p
m
is the largest power of
p
divides
n
, we find that
n
must be
even if p ≡ 3 (mod 4).
Conversely, let
n
=
p
n
1
1
p
n
2
2
···p
n
k
k
be a product of distinct primes. Now for
each p
i
, either p
i
≡ 3 (mod 4), and n
i
is even, in which case
p
n
i
i
= (p
2
i
)
n
i
/2
= N(p
n
i
/2
i
);
or
p
i
= 2 or
p
i
≡
1 (
mod
4), in which case, the above proof shows that
p
i
=
N
(
α
i
)
for some α
i
. So p
n
i
= N(α
n
i
).
Since the norm is multiplicative, we can write
n
as the norm of some
z ∈ Z
[
i
].
So
n = N(z) = N(x + iy) = x
2
+ y
2
,
as required.
Example. We know 65 = 5
×
13. Since 5
,
13
≡
1 (
mod
4), it is a sum of squares.
Moreover, the proof tells us how to find 65 as the sum of squares. We have to
factor 5 and 13 in Z[i]. We have
5 = (2 + i)(2 − i)
13 = (2 + 3i)(2 − 3i).
So we know
65 = N(2 + i)N(2 + 3i) = N((2 + i)(2 + 3i)) = N(1 + 8i) = 1
2
+ 8
2
.
But there is a choice here. We had to pick which factor is
α
and which is
¯α
. So
we can also write
65 = N((2 + i)(2 − 3i)) = N (7 − 4i) = 7
2
+ 4
2
.
So not only are we able to write them as sum of squares, but this also gives us
many ways of writing 65 as a sum of squares.