Iwasawa theory
The first thing Iwasawa noticed was that there are towers of fields in algebraic number theory, having Galois group isomorphic with the additive group of p-adic integers. That group, usually written Γ in the theory and with multiplicative notation, can be found as a subgroup of Galois groups of infinite field extensions (which are by their nature pro-finite groups). The group  is the inverse limit of the additive groups , where p is the fixed prime number and . We can express this by Pontryagin duality in another way: Γ is dual to the discrete group of all -power roots of unity in the complex numbers.