6 Auxiliary results
The ramification index of \(\mathfrak {P}\) above \(p\) is \(p - 1\), i.e.,
\[ e(\mathfrak {P}\mid p) = p - 1. \]
Proof
This follows from the general formula for the ramification index in cyclotomic extensions.
We have \(\zeta _p - 1 \notin \mathfrak {P}^2\).
Proof
By 105, \(e(\mathfrak {P}\mid p) = p - 1\). 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 \(\mathcal{O}_K/P\),
\[ g(1, \psi ) = \begin{cases} q - 1 & \text{if } \psi = 1 \\ -1 & \text{otherwise,} \end{cases} \]
where \(\psi = 1\) denotes the trivial character (identically equal to \(1\)).
Proof