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

TL;DR

This paper introduces new invariants for number fields that encode ramification data and fully determine the Dedekind zeta function, providing insights into the arithmetic structure of the field.

Contribution

It defines the $(eta,eta)$-ramification invariants and demonstrates their ability to characterize the Dedekind zeta function of a number field.

Findings

Invariants $eta_{p}^{L}$ are 1 for unramified primes.

Invariants $eta_{p}^{L}$ are divisible by $p$ for wildly ramified primes.

Residues of $eta_{p}^{L}$ mod $p$ determine the genus of the integral trace.

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.