Cyclotomic Integers: Unique Factorization Fails

Unique factorization fails for cyclotomic integers starting with λ = 23 where λ represent the primitive root of unity that a given cyclotomic integer is based on. For details on cyclotomic integers start here. This blog is part of the general proof that shows that Fermat's Last Theorem is true for regular primes.

The content is today's blog is based on Harold M. Edwards' Fermat's Last Theorem: A Genetic Introduction for Algebraic Number Theory. In today's blog, I take one result from Edwards. Edwards goes over a much more complete list of calculations taken from Ernst Kummer's original article.

Theorem 1: Not all cyclotomic integers are characterized by unique factorization

Proof:

(1) The method of this proof is to show that unique factorization fails for the number 47 when λ = 23.

NOTE: We only need to show one case of failure to show that unique factorization fails.

(2) Norm of (1 - α + α²¹) = 47*139.

NOTE: These calculations are taken directly from Edwards. If you see any mistake, please compare it to Edwards' book for the correction. Edwards takes his calculations from Kummer's original paper.

(a) Nf(α) = f(α)*f(α²)*...*f(α^λ-1) [Nf(α) = Norm of f(α) See Definition 2, here]

(b) Let G(α) = f(α)f(α⁴)f(α^-7)f(α^-5)f(α³)f(α^-11)f(α²)f(α⁸)f(α⁹)f(α^-10)f(α⁶)

where α^-x = α^23-x

(c) Then Nf(α) = G(α)G(α^-2)

since G(α^-2) = f(α^-2)f(α^-8)f(α¹⁴)f(α¹⁰)f(α^-6)f(α²²)f(α^-4)f(α^-16)f(α^-18)f(α²⁰)f(α^-12)

To be clear:

G(α) = f(α)f(α⁴)f(α¹⁶)f(α¹⁸)f(α³)f(α¹²)f(α²)f(α⁸)f(α⁹)f(α¹³)f(α⁶)

G(α^-2) = f(α²¹)f(α¹⁵)f(α¹⁴)f(α¹⁰)f(α¹⁷)f(α²²)f(α¹⁹)f(α⁷)f(α⁵)f(α²⁰)f(α¹¹)

(d) Let:

g(α) = f(α)f(α⁴)

h(α) = g(α)f(α^-7)

(e) Then:

G(α) = h(α)h(α^-5)h(α²)g(α^-10)

since this equals:

[f(α)f(α⁴)f(α¹⁶)][f(α¹⁸)f(α³)f(α¹²)][f(α²)f(α⁸)f(α⁹)][f(α¹³)f(α⁶)]

(f) Now, we can do the following computations:

g(α) = f(α)f(α⁴) = (1 - α + α²¹)(1 - α⁴ + α¹⁵) =
= α²¹ - α¹⁶ + α¹⁵ + α¹³ + α⁵ - α⁴ - α² - α + 1.

h(α) = g(α)f(α^-7) =
= (α²¹ - α¹⁶ + α¹⁵ + α¹³ + α⁵ - α⁴ - α² - α + 1)(1 - α¹⁶ + α¹⁰) =
= α²⁰ + α¹⁹ + α¹⁷ -3α¹⁶ + α¹³ + α¹² + α⁹ - α⁸ - α⁷ + α⁵ - α² - α + 1.

h(α^-5) = α¹⁵ + α²⁰ + α⁷ - 3α¹² + α⁴ + α⁹ + α - α⁶ - α¹¹ + α²¹ - α¹³ - α¹⁸ + 1.

h(α²) = α¹⁷ + α¹⁵ + α¹¹ - 3α⁹ + α³ + α + α¹⁸ - α¹⁶ - α¹⁴ + α¹⁰ - α⁴ - α² + 1

g(α^-10) = α²⁰ - α + α¹¹ + α⁸ + α¹⁹ - α⁶ - α³ - α¹³ + 1

(g) So, G(α) = -44 + 15θ₀ - 13θ₁

where θ₀ = α + α⁴ + α^-7 + ... + α⁶ and θ₁ = -1 - θ₀.

(h) Now, we can check it to find:

θ₀θ₁ = 11 + 5θ₀ + 5θ₁ = 6.

(i) Now, we are ready to put the calculation all together so that:

N(1 - α + α²¹) = G(α)G(α^-2) =

= (-31 + 28θ₀)(-31 + 28θ₁) = 31² - 31*28(θ₀ + θ₁) + 6*28² = 961 + 868 + 4704 = 6533 = 47*139.


(3) Norm(47) = 47*47*....*47 = 47²².

(4) Norm(139) = 139*139*...*139 = 139²²

(5) But, then (1 - α + α²¹) doesn't divide 47 and (1 - α + α²¹) doesn't divide 139 since (from Lemma 6, here), a cyclotomic only divides another cyclotomic integer if the norm of the first cyclotomic integer divides the norm of the second cyclotomic integer.

NOTE: In other words, 47*139 does not divide 139²² and likewise, 47*139 does not divide 47²².

(6) And this violates Euclid's Lemma since (1 - α + α²¹) does divide 47*139 by the equation in step #2 but it doesn't divide 47 and it doesn't divide 139.

NOTE: The equation proved in step #2 is:

N(1 - α + α²¹) = (1 - α + α²¹)*(1 - α² + α²¹)*...*(1 - α²¹ + α²¹) = 47*139.

(7) And this means that unique factorization does not exist in this situation. [See here for details on the Fundamental Theorem of Arithmetic and its dependence on Euclid's Lemma]

QED

Kummer then developed his theory of ideal numbers in order to save unique factorization for cyclotomic integers.