Are number fields determined by Artin L-Functions ?

TL;DR

This paper proves that Artin L-functions associated with faithful characters of Galois groups uniquely determine the Galois closure of number fields over the rationals, and in some cases, the characters themselves.

Contribution

It establishes that Artin L-functions encode complete information about the Galois closure of number fields and the characters in the case of rational base fields.

Findings

Artin L-functions determine the Galois closure over .

For , they also determine the character h.

The results link L-functions closely to field and Galois group structures.

Abstract

Let $k$ be a number field, $K/k$ a finite Galois extension with Galois group $G$, $\chi$ a faithful character of $G$. We prove that the Artin L-function $L(s,\chi,K/k)$ determines the Galois closure of $K$ over $\Q$. In the special case $k=\Q$ it also determines the character $\chi$.