The $(\alpha ,\beta )-$ramification invariants of a number field

Abstract

Let $L$ be a number field. For a given prime $p$ we define integers ${\alpha }_p^L$ and ${\beta }_p^L$ with some interesting arithmetic properties. For instance, ${\beta }_p^L$ is equal to $1$ whenever $p$ does not ramify in $L$ and ${\alpha }_p^L$ is divisible by $p$ whenever $p$ is wildly ramified in $L$. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of $L$. Moreover, if the residue class mod $p$ of ${\alpha }_p^L$ is not zero for all $p$ then such residues determine the genus of the integral trace.