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 formula_1 is the inverse limit of the additive groups formula_2, where "p" is the fixed prime number and formula_3. We can express this by Pontryagin duality in another way: Γ is dual to the discrete group of all formula_4-power roots of unity in the complex numbers.