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.
Today, I will continue reviewing the basic properties of cyclotomic integers. Today's content comes directly from Chapter 4 of Harold M. Edwards' Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory.
Lemma 1: Criteria for division of a cyclotomic integer by a rational integer.
A cyclotomic integer is divisible by a rational integer if and only if all of its integer coefficients are congruent modulo the rational integer.
(1) Let f(α) be a cyclotomic integer.
(2) f(α) = a0 + a1α + ... aλ-1αλ-1 [See Lemma 1 here for details]
(3) Let d be a rational integer.
(4) It is clear that if d divides f(α), then a0 ≡ a1 ≡ ... ≡ aλ-1 ≡ 0 (mod d).
(5) Assume a0 ≡ a1 ≡ ... ≡ aλ-1 (mod d).
(6) Then, using Corrolary 2.1 from here, we know that we can add aλ-1 to each of the coefficients to get:
f(α) = (a0 - aλ-1) + (a1 - aλ-1)α + ... + (aλ - 2 - aλ - 1)αλ-2 + (aλ-1 - aλ-1)αλ-1
(7) But then it is clear that d divides each of the coefficients so it divides f(α).
Corollary 1.1: Division Algorithm for a cyclotomic integer f(α) by a rational integer d
if f(α) = a0 + a1α + ... + aλ-1αλ-1, the result is:
[(a0 - aλ-1)/d] + [(a1 - aλ-1)/d]α + ... + [(aλ-2 - aλ-1)/d]αλ-2
(1) If d divides f(α), then a0 ≡ a1 ≡ ... ≡ aλ-1 (mod d) [From Lemma 1 above]
(2) By Corollary 2.1 here, we can add constant -c to each coefficient and still maintain the same value so that:
f(α) (a0 - aλ-1) + (a1 - aλ-1)α + ... + (aλ - 2 - aλ - 1)αλ-2 + (aλ-1 - aλ-1)αλ-1
(3) From this we know that d divides each coefficient and the result of this corrollary follows.
Lemma 2: Division Algorithm for Cyclotomic Integers
A cyclotomic integer h(α) is divisible by another cyclotomic integer f(α) if and only if:
h(α)*f(α)-1 is divisible by the Nf(α)
(1) Let f(α),h(α) be cyclotomic integers.
(2) We can assume f(α) ≠ 0
(a) Assume f(α) = 0.
(b) Then, g(α) exists only if h(α) = 0 in which case g(α) can take any value.
(c) This resolves f(α) = 0 so we only need to address the case where f(α) ≠ 0.
(3) Assume f(α) * g(α) = h(α)
(4) Then, Nf(α)*g(α) = h(α) *f(α2)*f(α3)*...*f(αλ-1) = h(α)*f(α)-1
(5) We see that f(α) divides h(α) if and only if Nf(α) divides h(α)*f(α3)*...*f(αλ-1).
(6) But Nf(α) is a rational integer. [See Lemma 5 here]
(7) Let i(α) = h(α)*f(α)-1
(8) i(α) = b0 + b1α + ... + bλ-1αλ-1
(9) Using Lemma 1 above, we see that Nf(α) divides i(α) if and only if for any two coefficients bj ≡ bk (mod Nf(α))
(10) But from step #5, we have that f(α) divides h(α) if and only if after computing i(α) from step #8, we find that all coeffients of i(α) are congruent modulo Nf(α).
Corollary 2.1: Method for determining result of division between two cyclotomic integers.
if f(α) = a0 + a1α + ... + aλ-1αλ-1 and:
h(α)f(α)-1 = b0 + b1α + ... + bλ-1αλ-1,
the result is:
[(b0 - bλ-1)/Nf(α)] + [(b1 - bλ-1)/Nf(α)]α + ... + [(bλ-2 - bλ-1)/Nf(α)]αλ-2
(1) Let f(α), g(α), h(α) be cyclotomic integers with h(α) = f(α)g(α).
(2) Multiplying both sides by f(α)-1 gives us:
Nf(α)g(α) = h(α)f(α)-1
(3) Since Nf(α) is a rational integer, we can apply Corollary 1.1 above to get the desired result.