Swan Modules and Hilbert–Speiser Number Fields, Original Research Article

A number field K is called a Hilbert–Speiser field if for each tamely ramified finite abelian extension N/K the ring of algebraic integers of N, imageN, has a normal integral basis over imageK, the ring of algebraic integers of K. The classical Hilbert–Speiser theorem proves that the field of rational numbers Q is such a field. It is well known that the class number of a Hilbert–Speiser field must equal 1. We consider tame elementary abelian extensions of a number field K and Swan modules to obtain additional necessary conditions for K to be a Hilbert–Speiser field. We show that among all number fields, the field Q is the only Hilbert–Speiser field.