Chebotarev Sets

TL;DR

This paper investigates conditions under which sets of primes or prime ideals can be expressed as finite unions of residue classes or Frobenius conjugacy classes, providing criteria and demonstrating limitations of such representations.

Contribution

It introduces criteria for representing prime sets as unions of residue classes or Frobenius classes and shows certain sets, like alternating primes, cannot be expressed this way.

Findings

Sets of primes can be characterized by specific criteria for such representations.

Not all prime sets, such as alternating primes, can be expressed as finite unions of residue classes.

The paper establishes limitations on representing prime sets in algebraic number theory.

Abstract

We consider the problem of determining whether a set of primes, or, more generally, prime ideals in a number field, can be realized as a finite union of residue classes, or of Frobenius conjugacy classes. We give criteria for a set to be realized in this manner, and show that the subset of primes consisting of every other prime cannot be expressed in this way, even if we allow a finite number of exceptions.