unique factorization of ideals

The fundamental theorem of arithmetic says every positive integer is a product of primes in essentially one way. When numbers in a number field stop factoring uniquely, all is not lost: the right objects to factor are ideals, not numbers, and at that level uniqueness comes roaring back. This is the cornerstone insight of the whole subject — factorization is restored by moving up to ideals.

The theorem states that in a Dedekind domain R, every nonzero proper ideal I can be written as a product of prime ideals I = P_1^{e_1} * P_2^{e_2} * ... * P_r^{e_r}, and this expression is unique up to the order of the factors. The same holds for nonzero fractional ideals, where the exponents may now be negative integers; equivalently the group of fractional ideals is free abelian on the set of nonzero prime ideals.

This both generalizes the fundamental theorem of arithmetic (R = Z, where prime ideals are just (p)) and explains the failure of unique factorization of elements: 6 = 2 * 3 = (1 + sqrt(-5))(1 - sqrt(-5)) are two factorizations into irreducible elements precisely because the prime ideals splitting 2 and 3 recombine into different principal ideals.