Introduction to Algebraic Integers — Extending the Notion of Integer

Using the Gaussian integers Z[i] and the Eisenstein integers Z[omega] as concrete examples, we introduce rings of algebraic integers. We demonstrate the failure of unique factorization in Z[sqrt(-5)] and show how ideals and prime ideal factorization restore uniqueness. We define the class number and discuss its significance.

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March 1, 2026

1 The Gaussian Integers

Definition 1 (Gaussian integers).

The ring of Gaussian integers is

Z [i] = {a + bi ∣ a, b \in Z}

, where

i = - 1

Definition 2 (Norm).

The norm of a Gaussian integer

α = a + bi

N (α) = a^{2} + b^{2} = α \overset{α}{ˉ}

, where

\overset{α}{ˉ} = a - bi

Theorem 3 (Multiplicativity of the norm).

N (α β) = N (α) N (β)

Proof.

N (α β) = α β \overline{α β} = α β \overset{α}{ˉ} \overset{ˉ}{β} = (α \overset{α}{ˉ}) (β \overset{ˉ}{β}) = N (α) N (β)

. □

Theorem 4 (

Z [i]

is a Euclidean domain).

Z [i]

is a Euclidean domain with respect to the norm

N

. That is, for any

α, β \in Z [i]

with

β \neq = 0

, there exist

γ, ρ \in Z [i]

such that

α = β γ + ρ

and

N (ρ) < N (β)

Proof.

Write

α / β = x + y i

with

x, y \in Q

. Choose integers

m, n

with

∣ x - m ∣ \leq 1/2

and

∣ y - n ∣ \leq 1/2

, and set

γ = m + ni

. Then

ρ = α - β γ

satisfies

N (ρ) = N (β) N (α / β - γ) = N (β) ((x - m)^{2} + (y - n)^{2}) \leq N (β) (1/4 + 1/4) = N (β) /2 < N (β)

. □

Corollary 5.

Z [i]

is a unique factorization domain (UFD).

Theorem 6 (Classification of Gaussian primes).

The primes (irreducible elements) of

Z [i]

, up to multiplication by units

{\pm 1, \pm i}

, fall into three types:

$1 + i$ (corresponding to the rational prime $2$ , since $N (1 + i) = 2$ ).
$a + bi$ where $N (a + bi) = p$ for a rational prime $p \equiv 1 (mod 4)$ .
$p$ , where $p$ is a rational prime with $p \equiv 3 (mod 4)$ .

2 The Eisenstein Integers

Definition 7 (Eisenstein integers).

Let

ω = e^{2 πi /3} = (- 1 + - 3) /2

. The ring of Eisenstein integers is

Z [ω] = {a + bω ∣ a, b \in Z}

, with norm

N (a + bω) = a^{2} - ab + b^{2}

Theorem 8.

Z [ω]

is a Euclidean domain and hence a UFD.

Remark 9.

The Eisenstein integers play an essential role in the proof of Fermat's last theorem for the exponent

n = 3

(showing that

x^{3} + y^{3} = z^{3}

has no nontrivial integer solutions).

3 The Failure of Unique Factorization

Theorem 10.

Z [- 5]

is not a UFD.

Proof.

Consider the norm

N (a + b - 5) = a^{2} + 5 b^{2}

Z [- 5]

. We have

6 = 2 \cdot 3 = (1 + - 5) (1 - - 5) .

The norms are

N (2) = 4

N (3) = 9

N (1 \pm - 5) = 6

. There is no

α = a + b - 5

with

N (α) = 2

(the equation

a^{2} + 5 b^{2} = 2

has no integer solutions), and similarly none with

N (α) = 3

. Therefore

2

3

1 + - 5

, and

1 - - 5

are all irreducible. Since

2

and

1 + - 5

are not associates (their norms differ), the integer

6

admits two essentially distinct irreducible factorizations. □

4 Ideals and Prime Ideal Factorization

Definition 11 (Ideal).

A nonempty subset

I

of a ring

R

is an ideal if

a - b \in I

for all

a, b \in I

and

r a \in I

for all

r \in R

a \in I

Theorem 12 (Dedekind's theorem).

Every nonzero ideal of the ring of integers

O_{K}

of an algebraic number field

K

can be written uniquely (up to the order of the factors) as a product of prime ideals.

Remark 13.

Dedekind's theorem guarantees that even when unique factorization of elements fails, unique factorization of ideals still holds. In

Z [- 5]

, although

6

does not factor uniquely into irreducible elements, at the level of ideals we have

(6) = (2, 1 + - 5)^{2} (3, 1 + - 5) (3, 1 - - 5)

, which is the unique prime ideal factorization.

Example 14.

Z [- 5]

(2) = p^{2}

where

p = (2, 1 + - 5)

, and

(3) = q \overset{ˉ}{q}

where

q = (3, 1 + - 5)

5 The Class Number

Definition 15 (Ideal class group and class number).

Let

I_{K}

denote the group of fractional ideals of

O_{K}

and

P_{K}

the subgroup of principal fractional ideals. The quotient group

Cl (K) = I_{K} / P_{K}

is called the ideal class group, and its order

h_{K} = ∣ Cl (K) ∣

is the class number.

Theorem 16.

O_{K}

is a UFD if and only if

h_{K} = 1

Proof.

h_{K} = 1

, then every ideal is principal, making

O_{K}

a principal ideal domain (PID). Every PID is a UFD (by a standard argument). Conversely, if

O_{K}

is a UFD, then every ideal is principal (this follows from properties of irreducible elements), so

h_{K} = 1

. □

Example 17.

h_{Q (i)} = 1

(so

Z [i]

is a UFD), while

h_{Q (- 5)} = 2

(so

Z [- 5]

is not a UFD). The imaginary quadratic fields

Q (d)

(

d < 0

) with class number

1

are finite in number: they correspond to

d = - 1, - 2, - 3, - 7, - 11, - 19, - 43, - 67, - 163

, a total of nine (by the theorem of Heegner–Stark–Baker).

Remark 18.

The class number is an important invariant that measures how far the ring of integers deviates from unique factorization. One can prove that the class number is always finite using the Minkowski bound, and concrete computations rely on the relationship between the discriminant of the field and the norms of ideals.

Number Theory Algebra Textbook Algebraic Integers Ideals

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Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

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Introduction to Algebraic Integers — Extending the Notion of Integer

Folio Official

March 1, 2026

1 The Gaussian Integers

Definition 1 (Gaussian integers).

The ring of Gaussian integers is

Z [i] = {a + bi ∣ a, b \in Z}

, where

i = - 1

Definition 2 (Norm).

The norm of a Gaussian integer

α = a + bi

N (α) = a^{2} + b^{2} = α \overset{α}{ˉ}

, where

\overset{α}{ˉ} = a - bi

Theorem 3 (Multiplicativity of the norm).

N (α β) = N (α) N (β)

Proof.

N (α β) = α β \overline{α β} = α β \overset{α}{ˉ} \overset{ˉ}{β} = (α \overset{α}{ˉ}) (β \overset{ˉ}{β}) = N (α) N (β)

. □

Theorem 4 (

Z [i]

is a Euclidean domain).

Z [i]

is a Euclidean domain with respect to the norm

N

. That is, for any

α, β \in Z [i]

with

β \neq = 0

, there exist

γ, ρ \in Z [i]

such that

α = β γ + ρ

and

N (ρ) < N (β)

Proof.

Write

α / β = x + y i

with

x, y \in Q

. Choose integers

m, n

with

∣ x - m ∣ \leq 1/2

and

∣ y - n ∣ \leq 1/2

, and set

γ = m + ni

. Then

ρ = α - β γ

satisfies

N (ρ) = N (β) N (α / β - γ) = N (β) ((x - m)^{2} + (y - n)^{2}) \leq N (β) (1/4 + 1/4) = N (β) /2 < N (β)

. □

Corollary 5.

Z [i]

is a unique factorization domain (UFD).

Theorem 6 (Classification of Gaussian primes).

The primes (irreducible elements) of

Z [i]

, up to multiplication by units

{\pm 1, \pm i}

, fall into three types:

$1 + i$ (corresponding to the rational prime $2$ , since $N (1 + i) = 2$ ).
$a + bi$ where $N (a + bi) = p$ for a rational prime $p \equiv 1 (mod 4)$ .
$p$ , where $p$ is a rational prime with $p \equiv 3 (mod 4)$ .

2 The Eisenstein Integers

Definition 7 (Eisenstein integers).

Let

ω = e^{2 πi /3} = (- 1 + - 3) /2

. The ring of Eisenstein integers is

Z [ω] = {a + bω ∣ a, b \in Z}

, with norm

N (a + bω) = a^{2} - ab + b^{2}

Theorem 8.

Z [ω]

is a Euclidean domain and hence a UFD.

Remark 9.

The Eisenstein integers play an essential role in the proof of Fermat's last theorem for the exponent

n = 3

(showing that

x^{3} + y^{3} = z^{3}

has no nontrivial integer solutions).

3 The Failure of Unique Factorization

Theorem 10.

Z [- 5]

is not a UFD.

Proof.

Consider the norm

N (a + b - 5) = a^{2} + 5 b^{2}

Z [- 5]

. We have

6 = 2 \cdot 3 = (1 + - 5) (1 - - 5) .

The norms are

N (2) = 4

N (3) = 9

N (1 \pm - 5) = 6

. There is no

α = a + b - 5

with

N (α) = 2

(the equation

a^{2} + 5 b^{2} = 2

has no integer solutions), and similarly none with

N (α) = 3

. Therefore

2

3

1 + - 5

, and

1 - - 5

are all irreducible. Since

2

and

1 + - 5

are not associates (their norms differ), the integer

6

admits two essentially distinct irreducible factorizations. □

4 Ideals and Prime Ideal Factorization

Definition 11 (Ideal).

A nonempty subset

I

of a ring

R

is an ideal if

a - b \in I

for all

a, b \in I

and

r a \in I

for all

r \in R

a \in I

Theorem 12 (Dedekind's theorem).

Every nonzero ideal of the ring of integers

O_{K}

of an algebraic number field

K

can be written uniquely (up to the order of the factors) as a product of prime ideals.

Remark 13.

Dedekind's theorem guarantees that even when unique factorization of elements fails, unique factorization of ideals still holds. In

Z [- 5]

, although

6

does not factor uniquely into irreducible elements, at the level of ideals we have

(6) = (2, 1 + - 5)^{2} (3, 1 + - 5) (3, 1 - - 5)

, which is the unique prime ideal factorization.

Example 14.

Z [- 5]

(2) = p^{2}

where

p = (2, 1 + - 5)

, and

(3) = q \overset{ˉ}{q}

where

q = (3, 1 + - 5)

5 The Class Number

Definition 15 (Ideal class group and class number).

Let

I_{K}

denote the group of fractional ideals of

O_{K}

and

P_{K}

the subgroup of principal fractional ideals. The quotient group

Cl (K) = I_{K} / P_{K}

is called the ideal class group, and its order

h_{K} = ∣ Cl (K) ∣

is the class number.

Theorem 16.

O_{K}

is a UFD if and only if

h_{K} = 1

Proof.

h_{K} = 1

, then every ideal is principal, making

O_{K}

a principal ideal domain (PID). Every PID is a UFD (by a standard argument). Conversely, if

O_{K}

is a UFD, then every ideal is principal (this follows from properties of irreducible elements), so

h_{K} = 1

. □

Example 17.

h_{Q (i)} = 1

(so

Z [i]

is a UFD), while

h_{Q (- 5)} = 2

(so

Z [- 5]

is not a UFD). The imaginary quadratic fields

Q (d)

(

d < 0

) with class number

1

are finite in number: they correspond to

d = - 1, - 2, - 3, - 7, - 11, - 19, - 43, - 67, - 163

, a total of nine (by the theorem of Heegner–Stark–Baker).

Remark 18.

Number Theory Algebra Textbook Algebraic Integers Ideals

Folio Official

Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

1 followers·107 articles

Introduction to Algebraic Integers — Extending the Notion of Integer

1 The Gaussian Integers

2 The Eisenstein Integers

3 The Failure of Unique Factorization

4 Ideals and Prime Ideal Factorization

5 The Class Number

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Introduction to Algebraic Integers — Extending the Notion of Integer

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2 The Eisenstein Integers

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5 The Class Number

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