Algebraic number theory - an illustrated guide | Is 5 a prime number?

Ideals generalize the concept of “all multiples of a number” to arbitrary rings. Rather than factoring individual numbers, algebraic number theory factors ideals, restoring unique factorization that fails at the element level.

Algebraic number theorists study number rings—generalizations of the integers to lattices in the complex plane. These systems reveal that fundamental properties like primality and unique factorization depend on the ambient number system.

Prime ideals generalize prime numbers to ideal theory, providing the irreducible building blocks for ideal factorization. These objects restore unique factorization at the ideal level in Dedekind domains.

The fundamental theorem of arithmetic states every integer factors uniquely into primes. This spectacularly fails in many number rings, revealing that unique factorization is special rather than universal.

In Dedekind domains (including all rings of integers in number fields), every nonzero ideal factors uniquely as a product of prime ideals. This theorem restores unique factorization lost at the element level.