MAT315: Introduction to Number Theory
Lecture 2: Fundamental Theorem of Arithmetic in the
Gaussian Integers
Malors Emilio Espinosa Lara
January 10, 2025
University of Toronto
The Gaussian Integers
Definition
Definition
A Gaussian Integer is a complex number a+ bi with a, b ∈ Z.
1
They form a grid
We can visualize them as forming a grid in the complex plane.
2
What made all work in Z?
What made all our work to function so far? That we were able to
divide one integer by another in a way that the residue strictly
decreased.
3
What made all work in Z?
Definition
Let z and w ̸= 0 be Gaussian Integers. We say z is divisible by w if
there exists a Gaussian Integer v such that
z = wv .
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What made all work in Z?
Example
Let z = 3 + 7i and w = −1 + 2i . Then
3 + 7i
−1 + 2i =
(3 + 7i)(−1− 2i)
(−1 + 2i)(−1− 2i)
=
11− 13i
5
=
11
5
− 13
5
i .
This last number is not a Gaussian Integer. So −1 + 2i does not divide
3 + 7i .
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What made all work in Z?
What made all our work to function so far? That we were able to
divide one integer by another in a way that the residue strictly
decreased.
In the complex numbers we cannot decide consistently who is bigger or
smaller.
We try to fix this by working with the square of its distance to the origin
(because it is always a nonnegative integer!).
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The norm and the circles
We define the norm of a Gaussian Integer as N(a+ bi) = a2 + b2. It is
the square of the usual complex norm N(z) = |z |2.
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Multiplicativity
Proposition
The norm is multiplicative, that is,
N((a+ bi)(x + yi)) = N(a+ bi)N(x + yi).
Proof.
Call z = x + yi . Then notice that
x2 + y2 = (x + yi)(x − yi) = zz = |z |2.
Then
N(zw) = |zw |2 = (|z ||w |)2 = |z |2|w |2 = N(z)N(w).
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How do we use the norm?
The norm connects the multiplicative structure of the Gaussian Integers
with that of the regular integers.
It allows us to translate problems in Gaussian Integers into problems of
regular Integers.
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How do we use the norm?
Proposition
The units of the Gaussian Integers are 1,−1, i ,−i
Proof.
Suppose u is a unit in the Gaussian Integers.. By definition it means
there exists v , in the Gaussian Integers, such that
uv = 1.
Now take the norm! We get
N(u)N(v) = N(1) = 1.
Hence, units must have norm 1! There are only four Gaussian integers
of norm 1 and by inspection all of them are units.
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How do we use the norm?
This means that every Gaussian Integer z has associated to it 4 trivial
multiples
z ,−z , iz ,−iz .
This is the analogous of a number n having the trivial multiples n and
−n.
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The grid of a number
The multiples form a grid of squares along the directions of the lines
through 0 and z , and the one 0 and iz .
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The grid of a number
The fundamental square has side length |z |.
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The grid of a number
Any Gaussian Integer is located in one of the squares (possibly in the
boundary).
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The Geometric Fact
For every point in a square there is one corner at distance strictly
smaller than the sidelength
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Division with remainder
Theorem
Let w , z be Gaussian Integers with z ̸= 0. Then there are Gaussian
integers q, r such that
w = zq + r ,
and with 0 ≤ N(r) < N(z).
Proof.
1. Locate w in the appropriate square of the grid of multiples of z .
2. zq is one of the closest corners.
3. r is the vector from this corner to w . It is a Gaussian Integer.
4. Its norm decreases by the above property of the square.
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Division with remainder
Example
Consider z = 3 + 2i and w = 7− 6i . The norms of these Gaussian
Integers are
N(3 + 2i) = 32 + 22 = 13,
N(−7− 6i) = (−7)2 + (−6)2 = 85.
Thus 7− 6i is the large one. So we divide by 3 + 2i .
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Division with remainder
Example
We create the grid of multiples of 3 + 2i . We notice that
−3(3 + 2i)
is the closest multiple to −7− 6i .
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Division with remainder
Example
The difference is the residue. We have
r = −7− 6i − (−3(3 + 2i)) = −7− 6i + 9 + 6i = 2.
Thus
−7− 6i = −3(3 + 2i) + 2,N(2) = 4 < 13 = N(3 + 2i).
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Fundamental Theorem of Arithmetic
We can now proceed in exactly the same way.
Definition
A Gaussian Integer is composite if it can be written as a product of two
non-unit Gaussian Integers.
A Gaussian Integer is prime if is not composite and is not a unit.
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Fundamental Theorem of Arithmetic
We now conclude, by the same process as for Z:
Theorem
Every Gaussian Integer that is not a unit has a unique prime factorization
up to units.
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