Iwasawa theory

Iwasawa theory is a branch of mathematics that studies how certain properties of algebraic objects, like number fields and their prime factorization patterns, change when you examine infinite towers of related fields. It provides a framework to understand the growth of class groups (which measure the failure of unique factorization) in these extensions, using tools from algebra and number theory. Essentially, Iwasawa theory helps mathematicians understand deep relationships between prime numbers, algebraic structures, and infinite sequences, revealing patterns that link local properties of primes to global number system behaviors.