Conductor-discriminant formula

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In mathematics, the conductor-discriminant formula or Führerdiskriminantenproduktformel, introduced by Hasse (1926, 1930) for abelian extensions and by Artin (1931) for Galois extensions, is a formula calculating the relative discriminant of a finite Galois extension L/K of local or global fields from the Artin conductors of the irreducible characters \mathrm{Irr}(G) of the Galois group G = G(L/K).

Statement[edit]

Let L/K be a finite Galois extension of global fields with Galois group G. Then the discriminant equals

\mathfrak{d}_{L/K} = \prod_{\chi \in \mathrm{Irr}(G)}\mathfrak{f}(\chi)^{\chi(1)},

where \mathfrak{f}(\chi) equals the global Artin conductor of \chi.[1]

Example[edit]

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}_p) = (\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 \mathfrak{f}_{(p)}(\chi) = (p^{\varphi(p^n)(n - 1/(p-1))}): the exponent is

\sum_{i=0}^{n-1}(\varphi(p^n) - \varphi(p^i)) = n\varphi(p^n) - 1 - (p-1)\sum_{i=0}^{n-2}p^i = n\varphi(p^n) - p^{n-1}.

Notes[edit]

  1. ^ Neukirch 1999, VII.11.9.

References[edit]