Let L = \mathbf{Q}(\zeta_{p^n})/\mathbf{Q} be a
cyclotomic extension of the rationals. The Galois group G equals (\mathbf{Z}/p^n)^\times. Because (p) is the only finite prime ramified, the global Artin conductor \mathfrak{f}(\chi) equals the local one \mathfrak{f}_{(p)}(\chi). Because G is abelian, every non-trivial irreducible character \chi is of degree 1 = \chi(1). Then, the local Artin conductor of \chi equals the conductor of the \mathfrak{p}-adic completion of L^\chi = L^{\mathrm{ker}(\chi)}/\mathbf{Q}, i.e. (p)^{n_p}, where n_p is the smallest natural number such that U_{\mathbf{Q}_p}^{(n_p)} \subseteq N_{L^\chi_\mathfrak{p}/\mathbf{Q}_p}(U_{L^\chi_\mathfrak{p}}). If p > 2, the Galois group G(L_\mathfrak{p}/\mathbf{Q}_p) = G(L/\mathbf{Q}) = (\mathbf{Z}/p^n)^\times is cyclic of order \varphi(p^n), and by
local class field theory and using that U_{\mathbf{Q}_p}/U^{(k)}_{\mathbf{Q}_p} = (\mathbf{Z}/p^k)^\times one sees easily that if \chi factors through a primitive character of (\mathbf{Z}/p^i)^\times, then \mathfrak{f}_{(p)}(\chi) = p^i whence as there are \varphi(p^i) - \varphi(p^{i-1}) primitive characters of (\mathbf{Z}/p^i)^\times we obtain from the formula \mathfrak{d}_{L/\mathbf{Q}} = (p^{\varphi(p^n)(n - 1/(p-1))}), the exponent is :: \sum_{i = 0}^{n} (\varphi(p^i) - \varphi(p^{i-1}))i = n\varphi(p^n) - 1 - (p-1)\sum_{i=0}^{n-2}p^i = n\varphi(p^n) - p^{n-1}. == Notes ==