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 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 ) . Iwasawa worked with so-called - extensions : infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of is of the form , so by Galois theory , a - extension is the same thing as a tower of fields such that .