How many algebraic integers of bounded height have a minimal polynomial with a given Galois group? One approach to this problem is via Malle's conjecture. In this talk we will discuss an alternative approach using a construction with the Galois group's group algebra which has proven fruitful in recent years. We will explain how this construction lets one apply tools from harmonic analysis and the theory of toric varieties. Applications include a Malle-free determination of the asymptotic count for cyclic Galois groups. Time permitting, we will also explain how this construction gives a new perspective on cubic abelian polynomials.