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

Ort: A3.339
Veranstalter: Prof. Dr. Jürgen Klüners

Titel: Fermat's Last Theorem for regular primes

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(ζp)), otherwise irregular.

Theorem. (Fermat’s Last Theorem for regular primes) For a regular prime p ≥ 3, the equation xp + yp = zp 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 | hp if and only if p divides the numerator of some of the Bernoulli numbers Bj
for j = 2, 4, , . . . , p − 3.

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

Theorem. p | hp 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 | hp.

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(ζp) and Q(ζp)+.