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Unlike ℤ and Gaussian integers, many algebraic number rings fail unique factorization: in ℤ[√−5], we have 6 = 2·3 = (1+√−5)(1−√−5) with non-associate factors. Ideal theory was developed to restore unique factorization by factoring into ideals instead of elements.
The Fundamental Theorem of Arithmetic tells you that every integer factors into primes uniquely — 12 = 2² × 3, and there is no other way to write it. The same is true for the Gaussian integers ℤ[i], which you showed is a Euclidean domain. But these happy cases disguise how special unique factorization actually is. Most algebraic number rings do not have it.
The classic example is ℤ[√−5], the ring of numbers a + b√−5 with a, b ∈ ℤ. In this ring, 6 factors in two genuinely different ways: 6 = 2 × 3, and also 6 = (1 + √−5)(1 − √−5). You can verify the second: (1 + √−5)(1 − √−5) = 1 − (−5) = 6. To confirm these are genuinely distinct factorizations and not just associate variants, you use the norm N(a + b√−5) = a² + 5b² to show that 2, 3, (1 + √−5), and (1 − √−5) are all irreducible in ℤ[√−5] but not primes. In ℤ, irreducible and prime coincide — but in rings without unique factorization, they come apart. An element p is prime if p | ab implies p | a or p | b; an element is irreducible if it cannot be written as a product of two non-units. In ℤ[√−5], the element 2 is irreducible (no element has norm 2) but not prime: 2 | (1 + √−5)(1 − √−5) = 6, yet 2 divides neither factor.
The resolution, due to Kummer and Dedekind, was to shift from factoring *elements* to factoring *ideals*. Even though the element 2 cannot be factored further in ℤ[√−5], the ideal (2) factors into a product of prime ideals: (2) = 𝔭² where 𝔭 = (2, 1 + √−5). When you re-express 6 = 2 × 3 = (1 + √−5)(1 − √−5) in terms of ideal factorizations, both sides yield the same product of prime ideals — uniqueness is restored at the level of ideals. The ideal class group measures how badly unique factorization fails for elements: if the class group is trivial, every ideal is principal and the ring is a UFD. The class group of ℤ[√−5] has order 2, encoding exactly the two-fold ambiguity. Algebraic number theory thus does not abandon unique factorization — it relocates it from elements to ideals.