## Ober­se­mi­nar "Num­ber Theo­ry and Arith­me­ti­cal Sta­ti­stics"

Ort: A3.339                                Uhrzeit: 14:15 - 15:45 Uhr

Das Seminar findet ab dem 09.04.2024 regelmäßig dienstags statt.

• Dienstag, 16.04.2024 Jan Diekmann "Fermat's Last Theorem for regular primes"
• Dienstag, 23.04.2024 Michael Baake, "Dynamical and spectral properties of some shift spaces of number-theoretic origin" (11:15 - 12:45 Uhr, Medienraum D2.314)
• Dienstag, 23.04.2024 Fabian Gundlach, Symmetries of the set of squarefree integers in a number field
• Dienstag, 21.05.2024 Marc Technau, The distribution of quadratic non-residues: A stroll through the garden
• Dienstag, 28.05.2024 Béranger Seguin, Algebraic Patching for Beginners
• Dienstag, 11.06.2024 Daniel Windisch, Algebraic geometry of equilibria in cooperative games

# 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

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(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)+.

# 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

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(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)+.