We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Let 
$F_{\pi }$ be a finite Galois-algebra extension of a number field F, with group G. Suppose that 
$F_{\pi }/F$ is weakly ramified and that the square root 
$A_\pi $ of the inverse different 
$\mathfrak {D}_{\pi }^{-1}$ is defined. (This latter condition holds if, for example, 
$|G|$ is odd.) Erez has conjectured that the class 
$(A_\pi )$ of 
$A_\pi $ in the locally free class group 
$\operatorname {\mathrm {Cl}}(\mathbf {Z} G)$ of 
$\mathbf {Z} G$ is equal to the Cassou–Noguès–Fröhlich root number class 
$W(F_{\pi }/F)$ associated with 
$F_\pi /F$. This conjecture has been verified in many cases. We establish a precise formula for 
$(A_\pi )$ in terms of 
$W(F_{\pi }/F)$ in all cases where 
$A_\pi $ is defined and 
$F_\pi /F$ is tame, and are thereby able to deduce that, in general, 
$(A_\pi )$ is not equal to 
$W(F_\pi /F)$.
