Kummer's theory of ideal numbers, developed by mathematician Ernst Eduard Kummer in the 19th century, addresses the limitations of traditional number theory when dealing with certain types of equations. It introduces the concept of "ideal numbers," which are not actual numbers but rather abstract entities that help in understanding the behavior of prime numbers in different number fields. This theory is particularly useful in the study of algebraic number theory and class field theory. By using ideal numbers, Kummer was able to resolve issues related to unique factorization in rings of integers, leading to significant advancements in the understanding of algebraic integers and their properties.