Adeles and Ideles

In the case of a locally compact field K, the additive group (K,+) and the multiplicative group (K×, ·) play key roles in the theory. Each is a locally compact abelian group, hence amenable to the methods of Fourier analysis. Moreover, the additive group is self-Pontrjagin dual, and the multiplicative group K× is a target group for class field theory on K: that is, there is a bijective correspondence between the finite abelian extensions L/K and the finite index open subgroups HL of K× such that K/HL ∼= Gal(L/K).