1 Setup
1.1 The cyclotomic field \(K\)
Let \(p\) be a prime number and \(f\) a positive integer. Set \(q = p^f\). Let \(K = \mathbb {Q}(\zeta _{q-1})\) be the \((q-1)\)-th cyclotomic field. We fix a maximal ideal \(P\) of \(\mathcal{O}_K\) lying above \(p\).
1.2 The Teichmüller character
The Teichmüller character is the unique group isomorphism
satisfying \(\omega (x) \equiv x \pmod{P}\) for all \(x \in (\mathcal{O}_K/P)^\times \), extended to all of \(\mathcal{O}_K/P\) by \(\omega (0) = 0\).
The order of \(\omega \) as a multiplicative character is \(q - 1\).
Since \(\omega \) is an isomorphism between groups of order \(q - 1\), it has order \(q - 1\) as a group homomorphism.
Let \(\sigma \) be a ring endomorphism of \(\mathcal{O}_L\) such that \(\sigma (\zeta _{q-1}) = \zeta _{q-1}^m\) for some \(m \in \mathbb {Z}\). Then \(\sigma (\omega (x)) = \omega (x)^m\) for all \(x \in \mathcal{O}_K/P\).
Since \(\omega (x) \in \mu _{q-1}(\mathcal{O}_K)\), we have \(\omega (x) = \zeta _{q-1}^k\) for some \(k\). Then \(\sigma (\omega (x)) = \sigma (\zeta _{q-1})^k = \zeta _{q-1}^{mk} = \omega (x)^m\).
For all \(a \in \mathbb {Z}\) with \((q-1) \nmid a\),
Since \((q-1) \nmid a\), the character \(\omega ^{-a}\) is nontrivial by 2. By MulChar.sum_eq_zero_of_ne_one, \(\sum _{x \in \mathcal{O}_K/P} \omega (x)^{-a} = 0\).
1.3 The additive character
Let \(L\) be an extension of \(K\) containing a primitive \(p\)-th root of unity \(\zeta _p\). We fix such a \(\zeta _p \in \mathcal{O}_L\).
The additive character \(\psi : \mathcal{O}_K/P \to \mu _p(\mathcal{O}_L)\) is defined by
where the exponent is lifted canonically to \(\mathbb {Z}\).
The character \(\psi \) is nontrivial, i.e., \(\psi \neq 1\).
Since the extension \(\mathcal{O}_K/P\) over \(\mathbb {Z}/p\mathbb {Z}\) is separable, the trace \(\mathrm{tr}: \mathcal{O}_K/P \to \mathbb {Z}/p\mathbb {Z}\) is nonzero, so there exists \(x\) with \(\mathrm{tr}(x) \neq 0\), hence \(\psi (x) = \zeta _p^{\mathrm{tr}(x)} \neq 1\).
The character \(\psi \) is primitive.
Since \(\mathcal{O}_K/P\) is a field, any nontrivial additive character is primitive. The character \(\psi \) is nontrivial by 6.
For all \(x \in \mathcal{O}_K/P\), \(\psi (x^p) = \psi (x)\).
The Frobenius map \(\varphi : x \mapsto x^p\) is a \((\mathbb {Z}/p\mathbb {Z})\)-algebra automorphism of \(\mathcal{O}_K/P\). By Algebra.trace_eq_of_algEquiv, \(\mathrm{tr}(\varphi (x)) = \mathrm{tr}(x)\), i.e., \(\mathrm{tr}(x^p) = \mathrm{tr}(x)\). Therefore \(\psi (x^p) = \zeta _p^{\mathrm{tr}(x^p)} = \zeta _p^{\mathrm{tr}(x)} = \psi (x)\).
Let \(\mathfrak {P}\) be a prime ideal of \(\mathcal{O}_L\) with \(\zeta _p - 1 \in \mathfrak {P}\). Then
for all \(x \in \mathcal{O}_K/P\).
There exists \(a \in \mathbb {N}\) such that \(\mathrm{tr}(x) = a\) and \(\psi (x) = \zeta _p^a\). By the binomial theorem,
Since \(\zeta _p - 1 \in \mathfrak {P}\), we have \((\zeta _p - 1)^n \in \mathfrak {P}^n \subseteq \mathfrak {P}^2\) for all \(n \geq 2\). Therefore
Let \(\mathfrak {P}\) be a prime ideal of \(\mathcal{O}_L\) with \(\zeta _p - 1 \in \mathfrak {P}\). Then \(\psi (x) \equiv 1 \pmod{\mathfrak {P}}\) for all \(x \in \mathcal{O}_K/P\).
By 9, \(\psi (x) \equiv 1 + \mathrm{tr}(x) \cdot (\zeta _p - 1) \pmod{\mathfrak {P}^2}\). Since \(\zeta _p - 1 \in \mathfrak {P}\), the term \(\mathrm{tr}(x) \cdot (\zeta _p - 1) \in \mathfrak {P}\), hence \(\psi (x) \equiv 1 \pmod{\mathfrak {P}}\).