Why did failures of unique factorization lead mathematicians to study algebraic integers and ideals?

In 19th-century number theory, mathematicians sought to extend the fundamental theorem of arithmetic—which guarantees that every natural number has a unique prime factorization—to broader domains of complex numbers, known as algebraic integers.12

However, these attempts revealed that unique factorization often fails in such domains. For example, in the domain of numbers of the form $a+b\sqrt{-5}$, the number 21 can be factored as both $3·7$ and $(4+\sqrt{-5})(4-\sqrt{-5})$. In this context, numbers like 3 are indecomposable but do not behave like traditional primes, as they fail to divide the products they appear in.1

To resolve these failures, Ernst Eduard Kummer introduced "ideal numbers," which were later refined by Richard Dedekind into the concept of ideals. By defining an ideal as a specific set of algebraic integers, Dedekind allowed the property of unique factorization to be restored, as these sets could be multiplied and factored uniquely. This shift toward abstraction and structural analysis became a cornerstone of modern algebra.342