Cyclotomic Integers: Periods

In today's blog, I will review the concept of periods for cyclotomic integers from Harold Edward's Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory. I am continuing down the same chapter that I started in an earlier blog. If you would like to start at the beginning of cyclotomic integer properties, start here.

I will continue to use Edward's σg(α) notation that I reviewed in a previous blog.

Lemma 1:

Let e be a factor of λ - 1 where λ is an odd prime and α is a primitive root of unity where α^λ = 1.

Then, for a given cyclotomic integer g(α) there exists a cyclotomic integer G(α) such that:

Ng(α) = G(α)*σG(α)*...*σ^e-1G(α)

Proof:

(1) Let G(α) = g(α)*σ^eg(α)*σ^2eg(α)*...*σ^-eg(α) where σ^-e denotes σ^λ-1-e

(2) With this definition we can see that:

G(α) = g(α)*g(α^γe)*g(α^γ2e)*...*g(α^γ(λ-1-e))

σG(α) = g(α^γ)*g(α^γ(e+1))*g(α^γ2e+1)*...*g(α^γλ-e)

...

σ^e-1G(α) = g(α^γ(e-1))*g(α^γ(2e-1))*...*g(α^γ(λ-2))

(3) Putting this all together, we see that:

Ng(α) = g(α)*g(α^γ)*g(α^γ*γ)*...*g(α^γ(λ-2))

(4) Since γ is a primitive root mod λ, we know that (3) is the same as:

Ng(α) = g(α)*g(α²)*g(α³)*..*g(α^λ-1)

QED

Example 1: λ = 13, γ = 2, and e = 4

In this case, Ng(α) = G(α)G(α)G(α⁴)G(α⁸)

Each term G(α) = g(α)*g(α²⁴)*g(α^22*4) = g(α)*g(α³)*g(α⁹)

Since:

2 ≡ 2 (mod 13)
2² ≡ 4 (mod 13)
2³ ≡ 8 (mod 13)
2⁴ ≡ 16 ≡ 3 (mod 13)
2⁵ ≡ 32 ≡ 6 (mod 13)
2⁶ ≡ 64 ≡ 12 (mod 13)
2⁷ ≡ 128 ≡ 11 (mod 13)
2⁸ ≡ 256 ≡ 9 (mod 13)
2⁹ ≡ 512 ≡ 5 (mod 13)
2¹⁰ ≡ 1024 ≡ 10 (mod 13)
2¹¹ ≡ 2048 ≡ 7 (mod 13)
2¹² ≡ 4096 ≡ 1 (mod 13)

Corollary 1.1: σ^eG(α) = G(α)

Proof:

(1) G(α) = g(α)*g(α^γe)*g(α^γ2e)*...*g(α^γ(λ-1-e))

(2) And we find that:

σ^eG(α) = g(α^γe)*g(α^γ2e)*g(α^γ3e)*...*g(α^γλ-1)

(3) And since γ^λ-1 ≡ 1 (mod λ) (By Fermat's Little Theorem), we see that the result in step #2 is the same as the result in step #1.

QED

Corollary 1.2: σ^eG(α) = G(α) implies that there exists a set of e cyclotomic integers: η₀, η₁, ... η_e-1 such that:

(a) η₀ = α + σ^eα + σ^2eα + ... + σ^λ-1-eα

(b) η_i+1 = ση_i

(c) G(α) = a₀ + a₁η₀ + a₂η₁ + ... + a_eη_e-1 where a_i are integers.

Proof:

(1) σ^eG(α) = G(α)

(2) Since G(α) is a cyclotomic integer, we know that (see Lemma 1 here):

G(α) = a₀ + a₁α + ... + a_λ-1α^λ-1

(3) Now, (#1) implies that the coefficient for any α^j, α^j = σ^eα^j

(4) In order for #3 to be true, then G(α) must have the following form:

G(α) = a₀ + a₁(α + σ^eα + σ^2eα + ... + σ^λ-1-e) + a₂(σα + σσ^eα + ...) + ... + a_e(σ^e-1α + σ^e-1σ^eα + ...)

The reason being that each time all the values G(α) shifts by e, the new value must also have had the same coefficient and further, all new resulting values must be the same as values that existed before the shift. For each coefficient, the set of possible α^j values must be closed.

(5) Now, we can define η₀ in the following way:

η₀ = α + σ^eα + σ^2eα + ... + σ^λ-1-e

(6) We can likewise define η_i+1 as:

η_i+1 = ση_i

(7) Putting (5) and (6) into the equation in step #4 gives us:

G(α) = a₀ + a₁η₀ + ... + a_eη_e-1

QED


Definition 1: Cyclotomic Period

Let λ be a prime integer. Let α be a primitive root of unity such that α^λ=1. Let γ be a primitive root modulo λ. Let e be a factor of λ - 1.

Let G(α) = g(α)*g(α^γe)*g(α^γ2e)*...*g(α^γ(λ-1-e))

Then there exists a set of e cyclotomic integers: η₀, η₁, ... η_e-1 such that:

(a) η₀ = α + σ^eα + σ^2eα + ... + σ^λ-1-eα

(b) η_i+1 = ση_i

(c) G(α) = a₀ + a₁η₀ + a₂η₁ + ... + a_eη_e-1 where a_i are integers.

The cyclotomic integers η₀, η₁, ... η_e-1 that meet conditions (a), (b), and (c) are called cyclotomic periods.

Note:

As a convention, η_e = η₀ and η_-1 = η_e-1. So one can think of η_i as an ever repeating pattern of e values such that η_i+1 = ση_i

In this way, η_i is defined for all integers i.

Example 1: period for λ = 13, μ = 2, γ = 4

In this case, the cyclotomic periods are:

η₀ = α + σ⁴α + σ⁸α = α + α³ + α⁹

η₁ = σα + σα³ + σα⁹ = α² + α⁶ + α⁵

η₂ = σ²α + σ²α³ + σ²α⁹ = α⁴ + α¹² + α¹⁰

η₃ = σ³α + σ³α³ + σ³α⁹ = α⁸ + α¹¹ + α⁷

Lemma 2: σ^eη_i = η_i

Proof:

(1) For Case 0, this is given by the note in definition 1 above.

σ^eη₀ = η_e = η₀

(2) Let's assume that this is true up to i so that:

σ^eη_i = η_i where 0 ≤ i.

(3) Then, this is also true for η_i+1 since:

σ^e(η_i+1) = σ^e(ση_{i) [From Definition 1 above]}

Now σ^eσ = σ^e+1 = σσ^e

So that we get:

σ^eη_i+1 = σ(σ^eη_i)

By assumption at step #2, σ^eη_i = η_i

So that we end up with:

σ^eη_i+1 = ση_i = η_i+1

(4) So, by the principle of induction, we are done.

QED


Lemma 3: Each period η consists of (λ-1)/e terms.

Proof:

(1) Let f = (λ - 1)/e.

(2) σ^efα = σ^λ-1α = α

(3) σ^{(λ - 1 - e)}α = σ^e(f-1)α = (σ^e)^f-1α

(4) So we can see that each term in α + σ^eα + σ^2eα + ... + σ^λ-1-eα is σⁱα where i is 0*e, 1*e, 2*e, ... , (f-1)*e.

QED

Definition 2: Made up of periods of length f

A cyclotomic integer is made up of periods of length f if it can be put in the form:

a₀ + a₁η₁ + a₂η₂ + ... + a_eη_e

where a_i are integers and η_i are periods of length f.

Lemma 4: A cyclotomic integer g(α) is made up of periods of length if and only if σ^eg(α) = g(α)

Proof:

(1) Assume that g(α) is a cyclotomic integer made up of periods of length f.

(2) Then, there exists a_i and η_i such that:

g(α) = a₀ + a₁η₀ + a₂η₁ + ... + a_eη_e-1

(3) Applying σ^e to g(α) gives us:

σ^eg(α) = a₀ + a₁σ^eη₀ + ... + a_eσ^eη_e-1

(4) Applying Lemma 2 above gives us:

σ^eg(α) = a₀ + a₁η₀ + ... + a_eη_e-1

(5) Assume that σ^eg(α) = g(α)

(6) Then by Corollary 1.2 above, g(α) consists of e periods.

(7) And by Lemma 3, each period will be of length f.

QED

Corollary 4.1: The product of two cyclotomic integers made up of periods of length f is itself made up of periods of length f.

Proof:

(1) Let f(α), g(α) be cyclotomic integers made up of periods of length f so that:

σ^ef(α) = f(α)
σ^eg(α) = g(α)

(2) But then:

σ^e[f(α)g(α)] = [σ^ef(α)][σ^eg(α)] = f(α)g(α)

(3) So that by Lemma 4 above, we have proved that f(α)g(α) must itself be made up of periods of length f.

QED

Example: λ = 13, f = 3, e = 4

We note that:

(η₀)² = η₁ + 2η₂

η₀η₁ = η₀ + η₁ + η₃

η₀η₂ = 3 + η₁ + η₃

Likewise, applying σ to these equations gives us:

(η₁)² = η₂ + 2η₀

η₁η₂ = η₁ + η₂ + η₀

And so forth.