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.
Categories: math.NT