The main object studied in harmonic analysis are locally compact (Abelian) groups. As several branches of number theory study locally compact fields, which are, in particular, locally compact groups, all the results on locally compact groups known in harmonic analysis are applicable in the study of locally compact fields.
A classical text on number theory that utilises harmonic analysis is André Weil's Basic Number Theory. The first chapter of this book is devoted to locally compact fields and utilises several results of harmonic analysis such as the existence and uniqueness of the Haar measure. It might be instructive to have a look into the book by Weil in order to see why harmonic analysis is a rather natural tool for studying locally compact fields.