6 Auxiliary results
Any abelian extension of \(\mathbb {Q}(\zeta _m)\) contained in \(\mathbb {Q}(\zeta _m, \zeta _p)\) and unramified at all primes is trivial.
Let \(F\) be a number field containing \(\mu _p\), let \(\mu \in F^\times \), and let \(L = F(\sqrt[p]{\mu })\). Let \(G_0 \leq \mathrm{Gal}(F/\mathbb {Q})\) be a subgroup. Then \(L/\mathbb {Q}\) is Galois with \(\mathrm{Gal}(L/\mathbb {Q})\) extending \(G_0\) if and only if for every \(\sigma \in G_0\) there exists \(\xi \in F^\times \) such that \(\sigma (\mu ) = \xi ^p \mu ^{a(\sigma )}\) for some \(a(\sigma ) \in \mathbb {Z}\). In particular, \(L/\mathbb {Q}\) is abelian if and only if for every \(\sigma _a \in \mathrm{Gal}(F/\mathbb {Q})\) there exists \(\xi \in F^\times \) such that \(\sigma _a(\mu ) = \xi ^p \mu ^a\).
The ramification index of \(\mathfrak {P}\) above \(p\) is \(p - 1\), i.e.,
This follows from the general formula for the ramification index in cyclotomic extensions.
We have \(\zeta _p - 1 \notin \mathfrak {P}^2\).
By 76, \(e(\mathfrak {P}\mid p) = p - 1\). Since \(\zeta _p\) is a primitive \(p\)-th root of unity, \((\zeta _p - 1)\) generates the unique prime above \(p\) in \(\mathbb {Q}(\zeta _p)\), which has ramification index \(p - 1\). Hence \(v_{\mathfrak {P}}(\zeta _p - 1) = 1\), so \(\zeta _p - 1 \notin \mathfrak {P}^2\).
For any additive character \(\psi \) of \(\mathcal{O}_K/P\),
where \(\psi = 1\) denotes the trivial character (identically equal to \(1\)).