Recenly I came across Peter Roquette's article *On the history of Artin's $L$-functions and conductors* (23 July 2003) in which he talks about some letters from Emil Artin and Emmy Noether to Helmut Hasse in the early 1930s.

Artin is trying to give the definitive form to the definition of his $L$-functions (to include ramified and archimedean places), and has proved what Hasse calls the *Führerdiskriminantenproduktformel* : for a finite galoisian extension $L|K$ of number fields with group $G=\mathrm{Gal}(L|K)$, the discriminant $\mathfrak{d}$ of $L|K$ can be decomposed as the product

$$ \prod_{\chi}\mathfrak{f}(\chi,L|K)^{\chi(1)} $$

extending over all characters $\chi$ of $G$, where $\mathfrak{f}(\chi,L|K)$ denotes the conductor of $\chi$ (as defined by Artin).

Emmy Noether writes to Hasse that she is looking for a decomposition formula for
the *diﬀerent* $\mathfrak{D}$ of $L|K$ which would yield Artin’s product formula for the
discriminant $\mathfrak{d}$ after applying the norm map $N_{L|K}$. Perhaps this is what she calls her *Zukunftsphantasie* (a fantasy for the future).

**Question**. Is there such a decomposition of the different $\mathfrak{D}$ ?