# L-series and isomorphisms of number fields

## TL;DR

This paper explores the relationship between number fields and their associated L-series, establishing conditions under which equal L-series imply isomorphism of the fields and character group isomorphisms.

## Contribution

It extends previous results by showing that isomorphisms between number fields correspond to L-series preserving isomorphisms between their character groups.

## Key findings

- Equal Dedekind zeta functions do not guarantee isomorphism.
- Equal sets of Dirichlet L-series imply field isomorphism.
- Isomorphisms between fields correspond to L-series preserving character group isomorphisms.

## Abstract

Two number fields with equal Dedekind zeta function are not necessarily isomorphic. However, if the number fields have equal sets of Dirichlet $L$-series then they \emph{are} isomorphic. We extend this result by showing that the isomorphisms between the number fields are in bijection with $L$-series preserving isomorphisms between the character groups.