Focusing on fruitful exchanges between group theory and number theory, this book examines recent work in the characterization of extensions of number fields in terms of the decomposition of prime ideals. A key problem in this area is establishing the equality of Dedekind zeta functions of different number fields. This problem was solved for abelian extensions by class field theory, but was little studied in its general form until 1970. Recent progress has been based on important results in group theory, particularly the complete classification of all finite simple groups. This book provides an overview of this progress in algebraic number theory; it contains previously unpublished work as well as numerous results appearing in monograph form the first time.