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(Sorry for my poor english...) Let $\chi$ be a Dirichlet character modulo $N$ and $\Psi_{\chi}$ be an one dimensional Galois representation such that

\begin{equation} \Psi_{\chi}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \text{Gal}(\mathbb{Q(\xi_{N}})/\mathbb{Q})\cong (\mathbb{Z}/N\mathbb{Z})^{*}\to \mathbb{C}^{*} \end{equation} where $\xi_N$ be the $N$-th root of unity. Then, I prove that the Artin conductor of $\Psi_{\chi}$ is the same as the conductor of $\chi$. However, I couldn't find the reference of this theorem. Please let me know if you have a reference to this theorem.

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  • $\begingroup$ If $N = p$ the character is ramified only at $p$ and $\Phi_p(x)$ is irreducible in $\Bbb{Q}_p$ and $v_p(\xi_p-1) = \frac{1}{\deg(\Phi_p)}v_p(\Phi_p(1)) = 1/(p-1)$ is totally ramified, $Gal(\Bbb{Q}_p(\xi_p)/\Bbb{Q}_p)=Gal(\Bbb{Q}(\xi_p)/\Bbb{Q})$ so the local Artin conductor is $f(\chi,p)=\sum_{i\ge 0}\frac{g_i}{g_0}(\chi(1)-\chi(G_i)) = \frac{g_0}{g_0}(1-0) = 1$ and the global conductor is $\prod_p p^{f(\chi,p)}=p$ $\endgroup$ – reuns Jul 8 at 14:09

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