# Cyclotomic root systems and bad primes

## TL;DR

This paper extends the concept of root systems to complex reflection groups, classifies them over number fields, and generalizes the notion of bad primes for spetsial groups, advancing algebraic understanding.

## Contribution

It introduces a generalized framework for root systems in complex reflection groups and classifies them over their fields of definition, also extending the concept of bad primes.

## Key findings

- Classification of root systems over number fields
- Generalization of bad primes for spetsial groups
- Framework for roots as modules over rings of integers

## Abstract

We generalize the definition and properties of root systems to complex reflection groups - roots become rank one projective modules over the ring of integers of a number field k. In the irreducible case, we provide a classification of root systems over the field of definition k of the reflection representation. In the case of spetsial reflection groups, we generalize as well the definition and properties of bad primes.