Decomposition of Primes in Torsion Point Fields

We investigate the decomposition of prime ideals in non-abelian extensions of number fields. These fields are generated by the coordinates of torsion points of elliptic curves without complex multiplications. We explain the necessary prerequisites from the theory of elliptic curves, modular forms, algebraic number theory, and invariant theory. Due to the complexity of the problem, complete results are restricted to torsion points of low order. These results are complemented by computational data which also cover some unsolved cases.