On a question of Perlis and Stuart regarding arithmetic equivalence

TL;DR

This paper investigates whether having identical Dedekind zeta functions for septic number fields implies that ramified primes share the same sum of ramification degrees, addressing a question posed by Perlis and Stuart.

Contribution

The paper provides a definitive answer to Perlis and Stuart's question for septic number fields, clarifying the relationship between zeta functions and ramification degrees.

Findings

For septic number fields, identical zeta functions do not necessarily imply equal sums of ramification degrees at ramified primes.

The study characterizes conditions under which the sums of ramification degrees coincide.

Counterexamples are constructed demonstrating the independence of zeta functions and ramification degree sums.

Abstract

Let $K$ be a number field. The $K$-arithmetic type of a rational prime $\ell$ is the tuple $A_{K}(\ell)=(f^{K}_{1},...,f^{K}_{g_{\ell}})$ of the residue degrees of $\ell$ in $K$, written in ascending order. A well known result of Perlis from the 70's states that two number fields have the same Dedekind zeta function if and only if for almost all primes $\ell$ the arithmetic types of $\ell$ in both fields coincide. By the end of the 90's Perlis and Stuart asked if having the same zeta function implies that for ramified primes the sum of the ramification degrees coincide. Here we study and answer their question for septic number fields.