Cyclotomic Integers: Factoring Fermat's Last Theorem

Today's blog continues the discussion of Kummer's proof of Fermat's Last Theorem for regular primes. If you would like to review the historical context for this proof, start here.

The major reason why cyclotomic integers are interesting in relation to Fermat's Last Theorem is because they enable us to factor Fermat's Last Theorem in the following way:

zⁿ = xⁿ + yⁿ = (x + y)(x + αy)(x + α²y) .... (x + α^n-1y)

Below I will show how I can derive this factoring using the Fundamental Theorem of Algebra.

Lemma 1: Let α be a primitive root of unity such that n is an odd prime and αⁿ = 1, and let x,y,z be integers such that xⁿ + yⁿ = zⁿ, then:

zⁿ = xⁿ + yⁿ = (x + y)(x + αy)(x + α²y) .... (x + α^n-1y)

Proof:

(1) We know that xⁿ - 1 has n root from the Fundamental Theorem of Algebra.

(2) We also note that for all αⁱ where 0 ≤ i ≤ n-1, we have (αⁱ)ⁿ = 1.

NOTE: αⁿ = 1 so it is really the same as α⁰.

(3) Based on #2, the Fundamental Theorem of Algebra gives us:

xⁿ - 1 = (x - 1)*(x - α)*(x - α²)*...*(x - α^n-1)

QED

Theorem 1: if n is odd, then zⁿ = xⁿ + yⁿ = (x + y)(x + αy)(x + α²y) .... (x + α^n-1y)

Proof:

(1) aⁿ - 1 = (a - 1)*(a - α)*(a - α²)*...*(a - α^n-1) [From Lemma 1 above]

(2) Since a can be any value, let a = -x/y so that:

(-x/y)ⁿ - 1 = [(-x/y) - 1]*[(-x/y) - α]*...*[(-x/y) - α^n-1] = -(x)ⁿ/yⁿ - 1

(3) If we multiply (-y)ⁿ=-(yⁿ) to both sides, we get:

xⁿ + yⁿ = (x + y)*(x + yα)*...*(x + α^n-1y)

QED