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 $F = Q (ζ_{p})$ be a unit such that

u \equiv a^{p} \mod λ_{p}^{p}

for some $a \in O_{F}$ . The either $u = ϵ^{p}$ for some $ϵ \in O_{F}^{\times}$ 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 $η \in K$ be as in 5.3. Then $K / F$ is Galois and cyclic of degree $p$ . Now by Hilbert 90 (5.2) we can find $β \in O_{K}$ such that

η = β / σ (β)

(but note that by our assumption $β$ is not a unit). Note that the ideal $(η)$ is invariant under $Gal (K / F)$ by construction and by 5.9 it cannot be ramified, therefore $(β)$ is the extension of scalars of some ideal $b$ in $O_{F}$ . Now if $b$ is principal generated by some $γ$ then $β = v γ$ for some $v \in O_{K}^{\times}$ . But this means that

η = β / σ (β) = v / (σ (v)) \cdot γ / σ (γ) = v / σ (v)

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