Conductor-discriminant formula

In mathematics, the conductor-discriminant formula is a formula calculating the discriminant of a finite Galois extension L / K of global fields from the global Artin conductors of the irreducible characters Irr(G) of the Galois group G = G(L / K).

Statement

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 χ.^[1]

Example

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 χ is of degree 1 = χ(1). Then, the local Artin conductor of χ 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 φ(pⁿ), 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

1. ^ Neukirch 1999, VI.11.9.

References

• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, MR1697859