Fermat’s Last Theorem for regular primes: Proof of Kummers Lemma

5.10 Proof of Kummers Lemma

Using the above we have the following lemma (from which Kummer’s lemma is immediate):

Lemma 5.11 ✓

Let \(u \in F\) with \(F=\mathbb {Q}(\zeta _p)\) be a unit such that

\[ u \equiv a^p \mod \lambda _p^p \]

for some \(a \in \mathcal{O}_F\). The either \(u = \epsilon ^p\) for some \(\epsilon \in \mathcal{O}_F^\times \) or \(p\) divides the class number of \(F\).

Proof ▼

Assume \(u\) is not a \(p\)-th power. Then let \(K=F(\sqrt[p]{u})\) and \(\eta \in K\) be as in 5.3. Then \(K/F\) is Galois and cyclic of degree \(p\). Now by Hilbert 90 (5.2) we can find \(\beta \in \mathcal{O}_K\) such that

\[ \eta = \beta /\sigma (\beta ) \]

(but note that by our assumption \(\beta \) is not a unit). Note that the ideal \((\eta )\) is invariant under \(\operatorname{Gal}(K/F)\) by construction and by 5.9 it cannot be ramified, therefore \((\beta )\) is the extension of scalars of some ideal \(\mathfrak {b}\) in \(\mathcal{O}_F\). Now if \(\mathfrak {b}\) is principal generated by some \(\gamma \) then \(\beta =v \gamma \) for some \(v \in \mathcal{O}_K^\times \). But this means that

\[ \eta = \beta /\sigma (\beta )= v/(\sigma (v)) \cdot \gamma / \sigma (\gamma )=v/\sigma (v) \]

contradicting our assumption coming from 5.3. On the other hand, \(\mathfrak {b}^p= N_{K/F}(\beta )\) is principal, meaning \(p\) divides the class group of \(F\) as required.