# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1906.04254