Oberseminar "Number Theory and Arithmetical Statistics": Jan Diekmann (Duisburg-Essen), Fermat's Last Theorem for regular primes

Titel: Fermat's Last Theorem for regular primes

Abstract:
Definition. For a number field K, let h(K) denote its class number. A prime number p is said to be regular if p ∤ h(Q(ζ_p)), otherwise irregular.

Theorem. (Fermat’s Last Theorem for regular primes) For a regular prime p ≥ 3, the equation x^p + y^p = z^p does not have a solution in positive integers.

The proof of this theorem is divided into two cases: p ∤ xyz and p | xyz. The latter of these two cases is more complex than the first one. It uses a certain result called Kummer’s Lemma. The proof of Kummer’s Lemma relies on global class field theory.

Theorem. (Kummer’s criterion) Let p be some prime number. Then p | h_p if and only if p divides the numerator of some of the Bernoulli numbers B_j
for j = 2, 4, , . . . , p − 3.

The maximal totally real subfield of Q(ζ_p), Q(ζ_p)⁺ = Q(ζ_p + ζ_p⁻¹ ) satisfies h(Q(ζ_p)⁺) =: h⁺_p | h_p := h(Q(ζ_p)) (proved via global class field theory). We then define h⁻_p := h_p/h⁺_p . From this point, the proof of Kummer’s criterion consists of the following two theorems:

Theorem. p | h⁻_p if and only if p divides the numerator of the Bernoulli number Bj for some j = 2, 4, . . . , p − 3.

Theorem. If p | h⁺_p then p | h⁻_p.

The proof of this result relies on global class field theory and on a certain Galois module structure of the p-Sylow subgroups of the ideal class groups of Q(ζ_p) and Q(ζ_p)⁺.