2 Third Cyclotomic Extensions

Theorem 2.1

✓

Let $K = Q (ζ_{3})$ be the third cyclotomic field.
Let $O_{K} = Z [ζ_{3}]$ be the ring of integers of $K$ .
Let $O_{K}^{\times}$ be the group of units of $O_{K}$ .
Let $ζ_{3} \in K$ be any primitive third root of unity.
Let $η \in O_{K}$ be the element corresponding to $ζ_{3} \in K$ .
Let $λ \in O_{K}$ be such that $λ = η - 1$ .
Let $u \in O_{K}^{\times}$ be a unit.

Then $u \in {1, - 1, η, - η, η^{2}, - η^{2}}$ .

Proof ▶

Let $F$ be the fundamental system of $K$ .
By properties of cyclotomic fields, we know that $rank (K) = 0$ (see this lemma, this lemma and this lemma which have already been formalised and included in Mathlib). By the Dirichlet Unit Theorem (see Mathlib), we know that

\exists x \in K with finite order, such that u = x \prod_{v \in F} v,

but since $rank (K) = 0$ , then $F = \emptyset$ , which implies that $u = x$ .
Since $u = x$ has finite order, by properties of primitive roots (see this lemma that has already been formalised and included in Mathlib), we can deduce that

\exists r < 3 such that u = η^{r} \lor u = - η^{r} .

Therefore, we can conclude

u \in {\pm η^{r} ∣ r \in {0, 1, 2}} = {1, - 1, η, - η, η^{2}, - η^{2}} .

Theorem 2.2

✓

Proof ▶

By properties of cyclotomic fields, we know that ${1, η}$ is an integral power basis of $O_{K}$ (see this lemma, this lemma and this lemma which have already been formalised and included in Mathlib).
For every $ξ \in O_{K}$ , we define $π_{1} (ξ)$ and $π_{2} (ξ)$ to be the first and second coordinates of $ξ$ with respect to the basis ${1, η} \in O_{K}$ , i.e.

ξ = π_{1} (ξ) + π_{2} (ξ) η .

By contradiction we assume that

\exists m \in Z such that 3 ∣ η - m,

which implies that

\exists x \in O_{K} such that η - m = 3 x .

By linearity of $π_{2}$ ,

π_{2} (η) = π_{2} (3 x + m) = 3 π_{2} (x) + π_{2} (m) .

Since $π_{2} (η) = 1$ and $π_{2} (m) = 0$ , then we have that $3 ∣ 1$ , which is a contradiction.

Lemma 2.3

✓

Proof ▶

By definition we have that $λ = η - 1$ , which implies that

λ^{2} = (η - 1)^{2} = η^{2} - 2 η + 1.

Since $η$ corresponds to a root of the equation $x^{2} + x + 1 = 0$ , then $η^{2} = - 1 - η$ . Substituting back, we can conclude that

λ^{2} = (- 1 - η) - 2 η + 1 = - 3 η .

Theorem 2.4

✓

Let $K = Q (ζ_{3})$ be the third cyclotomic field.
Let $O_{K} = Z [ζ_{3}]$ be the ring of integers of $K$ .
Let $O_{K}^{\times}$ be the group of units of $O_{K}$ .
Let $ζ_{3} \in K$ be any primitive third root of unity.
Let $η \in O_{K}$ be the element corresponding to $ζ_{3} \in K$ .
Let $λ \in O_{K}$ be such that $λ = η - 1$ .
Let $u \in O_{K}^{\times}$ be a unit.

If $\exists m \in Z$ such that $λ^{2} ∣ u - m$ , then $u = 1 \lor u = - 1$ .
This is a special case of the Kummer’s Lemma.

Proof ▶

By Lemma 2.3, we have that $- 3 η = λ^{2} ∣ u - m$ , which implies that $3 ∣ u - m$ .
By Theorem 2.1, we know that $u \in {1, - 1, η, - η, η^{2}, - η^{2}}$ .
We proceed by analysing each case:

Case $u = 1 \lor u = - 1$ . This finishes the proof.
Case $u = η$ .
Since $3 ∣ u - m$ , we have that $3 ∣ η - m$ , which contradicts Theorem 2.2 forcing us to conclude that $u \neq η$ .
Case $u = - η$ .
Since $3 ∣ u - m$ , we have that $3 ∣ - η - m$ , then by properties of divisibility $3 ∣ η + m$ , which contradicts Theorem 2.2 forcing us to conclude that $u \neq - η$ .
Case $u = η^{2}$ .
Since $3 ∣ u - m$ , we have that $3 ∣ η^{2} - m$ , which contradicts Theorem 2.2 since $η^{2}$ is a third root of unity (see Mathlib), forcing us to conclude that $u \neq η^{2}$ .
Case $u = - η^{2}$ .
Since $3 ∣ u - m$ , we have that $3 ∣ - η^{2} - m$ , then by properties of divisibility $3 ∣ η^{2} + m$ , which contradicts Theorem 2.2 since $η^{2}$ is a third root of unity (see Mathlib), forcing us to conclude that $u \neq - η^{2}$ .

Therefore, $u = 1 \lor u = - 1$ .

Lemma 2.5

✓

Proof ▶

Since the third cyclotomic polynomial over $Q$ is irreducible, then the norm of $λ$ is $3$ by properties of primitive roots (see this lemma that has already been formalised and included in Mathlib).

Lemma 2.6

✓

Proof ▶

It directly follows from Lemma 2.5 since $3$ is a prime number.

Lemma 2.7

✓

Proof ▶

By properties of norms and divisibility, if the norm of an element in the ring of integers divides a number, then the element itself must divide that number. In this case, by Lemma 2.5 we know that the norm of $λ$ is $3$ , that divides $3$ , which implies that $λ ∣ 3$ .

Lemma 2.8

✓

Proof ▶

Since $3$ is prime and $ζ_{3}$ is a primitive third root of unity, then $λ$ is prime by Theorem 1.29.

Lemma 2.9

✓

Proof ▶

Lemma 2.10

✓

Proof ▶

Lemma 2.11

✓

Let $K = Q (ζ_{3})$ be the third cyclotomic field.
Let $O_{K} = Z [ζ_{3}]$ be the ring of integers of $K$ .
Let $O_{K}^{\times}$ be the group of units of $O_{K}$ .
Let $ζ_{3} \in K$ be any primitive third root of unity.
Let $η \in O_{K}$ be the element corresponding to $ζ_{3} \in K$ .
Let $λ \in O_{K}$ be such that $λ = η - 1$ .
Let $I$ be the ideal generated by $λ$ .

Then $O_{K} / I$ has cardinality $3$ .

Proof ▶

Lemma 2.12

✓

Let $K = Q (ζ_{3})$ be the third cyclotomic field.
Let $O_{K} = Z [ζ_{3}]$ be the ring of integers of $K$ .
Let $O_{K}^{\times}$ be the group of units of $O_{K}$ .
Let $ζ_{3} \in K$ be any primitive third root of unity.
Let $η \in O_{K}$ be the element corresponding to $ζ_{3} \in K$ .
Let $λ \in O_{K}$ be such that $λ = η - 1$ .
Let $I$ be the ideal generated by $λ$ .
Let $2 \in O_{K} / I$ .

Then $2 \neq 0$ .

Proof ▶

By contradiction we assume that $2 \in I$ , then, by definition, $λ$ would divide $2 \in O_{K}$ . Recall from Lemma 2.5 that the norm of $λ$ is $3$ . If $λ$ divided $2$ , then by properties of divisibility in number fields, the norm of $λ$ would also divide $2$ . However $3 ∤ 2$ showing a contradiction. Therefore, $λ ∤ 2$ , then $2 \notin I$ , which implies that $2 \in O_{K} / I$ is non-zero.

Lemma 2.13

✓

Proof ▶

By contradiction we assume that $λ ∣ 2$ , that implies that $2 \in I$ from which it follows that $2 = 0$ contradicting Lemma 2.12 forcing us to conclude that $λ ∤ 2$ .

Lemma 2.14

✓

Let $K = Q (ζ_{3})$ be the third cyclotomic field.
Let $O_{K} = Z [ζ_{3}]$ be the ring of integers of $K$ .
Let $O_{K}^{\times}$ be the group of units of $O_{K}$ .
Let $ζ_{3} \in K$ be any primitive third root of unity.
Let $η \in O_{K}$ be the element corresponding to $ζ_{3} \in K$ .
Let $λ \in O_{K}$ be such that $λ = η - 1$ .
Let $I$ be the ideal generated by $λ$ .

Then $O_{K} / I = {0, 1, - 1}$ .

Proof ▶

By Lemma 2.11, the cardinality of $O_{K} / I$ is $3$ , so it suffices to prove that $1, - 1$ and $0$ are distinct.
We proceed by contradiction analysing each case:

Case $1 = - 1$ . By basic algebraic properties, $1 = - 1$ implies that $2 = 0$ , which contradicts Lemma 2.12 forcing us to conclude that $1 \neq - 1$ .
Case $1 = 0$ . Trivial contradiction.
Case $- 1 = 0$ . It implies that $1 = 0$ , which is a contradiction.

Lemma 2.15

✓

Let $K = Q (ζ_{3})$ be the third cyclotomic field.
Let $O_{K} = Z [ζ_{3}]$ be the ring of integers of $K$ .
Let $O_{K}^{\times}$ be the group of units of $O_{K}$ .
Let $ζ_{3} \in K$ be any primitive third root of unity.
Let $η \in O_{K}$ be the element corresponding to $ζ_{3} \in K$ .
Let $λ \in O_{K}$ be such that $λ = η - 1$ .
Let $x \in O_{K}$ .

Then $(λ ∣ x) \lor (λ ∣ x - 1) \lor (λ ∣ x + 1)$ .

Proof ▶

Let $I$ be the ideal generated by $λ$ . Let $π : O_{K} \to O_{K} / I$ .
By Lemma 2.14, we have that $π (x) \in O_{K} / I = {0, 1, - 1}$ .
We proceed by analysing each case:

Case $π (x) = 0$ . By properties of ideals, $λ ∣ x$ .
Case $π (x) = 1$ . Then $0 = π (x) - 1 = π (x - 1)$ , which, by properties of ideals, implies that $λ ∣ x - 1$ .
Case $π (x) = - 1$ . Then $0 = π (x) + 1 = π (x + 1)$ , which, by properties of ideals, implies that $λ ∣ x + 1$ .

Lemma 2.16

✓

Proof ▶

Since $ζ_{3} \in K$ is a primitive third root of unity, then $ζ_{3}^{3} = 1$ . Given that $η \in O_{K}$ is the element corresponding to $ζ_{3} \in K$ , then $η^{3} = 1$ by the extension of the field properties into the ring of integers.

Lemma 2.17

✓

Proof ▶

Lemma 2.18

✓

Proof ▶

Since $η$ corresponds to a root of the equation $x^{2} + x + 1 = 0$ , then $η^{2} + η + 1 = 0$ .

Lemma 2.19

✓

Let $K = Q (ζ_{3})$ be the third cyclotomic field.
Let $O_{K} = Z [ζ_{3}]$ be the ring of integers of $K$ .
Let $O_{K}^{\times}$ be the group of units of $O_{K}$ .
Let $ζ_{3} \in K$ be any primitive third root of unity.
Let $η \in O_{K}$ be the element corresponding to $ζ_{3} \in K$ .
Let $x \in O_{K}$ .

Then $x^{3} - 1 = (x - 1) (x - η) (x - η^{2})$ .

Proof ▶

Applying Lemma 2.16 and Lemma 2.18, we have that

\begin{aligned} (x - 1) (x - η) (x - η^{2}) & = x^{3} - x^{2} (η^{2} + η + 1) + x (η^{2} + η + η^{3}) - η^{3} \\ = x^{3} - x^{2} (η^{2} + η + 1) + x (η^{2} + η + 1) - 1 \\ = x^{3} - 1. \end{aligned}

Lemma 2.20

✓

Let $K = Q (ζ_{3})$ be the third cyclotomic field.
Let $O_{K} = Z [ζ_{3}]$ be the ring of integers of $K$ .
Let $O_{K}^{\times}$ be the group of units of $O_{K}$ .
Let $ζ_{3} \in K$ be any primitive third root of unity.
Let $η \in O_{K}$ be the element corresponding to $ζ_{3} \in K$ .
Let $λ \in O_{K}$ be such that $λ = η - 1$ .
Let $x \in O_{K}$ .

Then $λ ∣ x (x - 1) (x - (η + 1))$ .

Proof ▶

By Lemma 2.15, we have that

(λ ∣ x) \lor (λ ∣ x - 1) \lor (λ ∣ x + 1) .

We proceed by analysing each case:

Case $λ ∣ x$ .
By properties of divisibility, we have that $λ ∣ x (x - 1) (x - (η + 1))$ .
Case $λ ∣ x - 1$ .
By properties of divisibility, we have that $λ ∣ x (x - 1) (x - (η + 1))$ .
Case $λ ∣ x + 1$ .
By properties of divisibility, it suffices to prove that

$λ ∣ x - (η + 1) = x + 1 - (η - 1 + 3) .$

By definition of $λ$ , we have that

$x + 1 - (η - 1 + 3) = x + 1 - (λ + 3) .$

By properties of divisibility and Lemma 2.7, we can deduce that $λ ∣ λ + 3$ .
Therefore, by properties of divisibility, we can conclude that

$λ ∣ x (x - 1) (x - (η + 1)) .$

Lemma 2.21

✓

Let $K = Q (ζ_{3})$ be the third cyclotomic field.
Let $O_{K} = Z [ζ_{3}]$ be the ring of integers of $K$ .
Let $O_{K}^{\times}$ be the group of units of $O_{K}$ .
Let $ζ_{3} \in K$ be any primitive third root of unity.
Let $η \in O_{K}$ be the element corresponding to $ζ_{3} \in K$ .
Let $λ \in O_{K}$ be such that $λ = η - 1$ .
Let $x \in O_{K}$ .

If $λ ∣ x - 1$ , then $λ^{4} ∣ x^{3} - 1$ .

Proof ▶

Let $λ ∣ x - 1$ , which is equivalent to say that

\exists y \in O_{K} such that x - 1 = λ y .

By ring properties and Lemma 2.19, we have that

x^{3} - 1 = λ^{3} (y (y - 1) (y - (η + 1))) .

By properties of divisibility and Lemma 2.20, we can conclude that

λ^{4} ∣ x^{3} - 1.

Lemma 2.22

✓

Let $K = Q (ζ_{3})$ be the third cyclotomic field.
Let $O_{K} = Z [ζ_{3}]$ be the ring of integers of $K$ .
Let $O_{K}^{\times}$ be the group of units of $O_{K}$ .
Let $ζ_{3} \in K$ be any primitive third root of unity.
Let $η \in O_{K}$ be the element corresponding to $ζ_{3} \in K$ .
Let $λ \in O_{K}$ be such that $λ = η - 1$ .
Let $x \in O_{K}$ .

If $λ ∣ x + 1$ , then $λ^{4} ∣ x^{3} + 1$ .

Proof ▶

By properties of divisibility, if $λ ∣ x + 1$ then

λ ∣ - (x + 1) = (- x) - 1.

By Lemma 2.20, we can deduce that

λ^{4} ∣ (- x)^{3} - 1.

By divisibility and ring properties we can conclude that

λ^{4} ∣ x^{3} + 1.

Lemma 2.23

✓

Let $K = Q (ζ_{3})$ be the third cyclotomic field.
Let $O_{K} = Z [ζ_{3}]$ be the ring of integers of $K$ .
Let $O_{K}^{\times}$ be the group of units of $O_{K}$ .
Let $ζ_{3} \in K$ be any primitive third root of unity.
Let $η \in O_{K}$ be the element corresponding to $ζ_{3} \in K$ .
Let $λ \in O_{K}$ be such that $λ = η - 1$ .
Let $x \in O_{K}$ .

If $λ ∤ x$ , then $(λ^{4} ∣ x^{3} - 1) \lor (λ^{4} ∣ x^{3} + 1)$ .

Proof ▶

By Lemma 2.15, we have that

(λ ∣ x) \lor (λ ∣ x - 1) \lor (λ ∣ x + 1) .

We proceed by analysing each case:

Case $λ ∣ x$ . From trivially contradictory hypotheses we can conclude that

$(λ^{4} ∣ x^{3} - 1) \lor (λ^{4} ∣ x^{3} + 1) .$
Case $λ ∣ x - 1$ . By Lemma 2.21, we have that $λ^{4} ∣ x^{3} - 1$ , which implies that

$(λ^{4} ∣ x^{3} - 1) \lor (λ^{4} ∣ x^{3} + 1) .$
Case $λ ∣ x + 1$ . By Lemma 2.22, we have that $λ^{4} ∣ x^{3} + 1$ , which implies that □

$(λ^{4} ∣ x^{3} - 1) \lor (λ^{4} ∣ x^{3} + 1) .$