Dedekind domains and ideal factorization

A Dedekind domain is a Noetherian integrally closed integral domain in which every nonzero prime ideal is maximal, with the field case often treated separately. The full ring of integers of a number field is a nonfield example. Its localization at any nonzero prime ideal is a discrete valuation ring. These local rings measure the exponent of that prime in a nonzero ideal, while the global theorem asserts that only finitely many exponents occur for each integral ideal.

Unique factorization applies to nonzero ideals; the zero ideal is excluded. The unit ideal has the empty factorization. Every nonzero fractional ideal is invertible, allowing integer exponents and forming an abelian group generated freely by nonzero prime ideals. Multiplication of ideals consists of finite sums of products of elements, not simply a union or intersection of their element sets. Containment reverses the intuitive divisibility direction: an ideal with a larger prime exponent is a smaller subset.

An irreducible element cannot be written as a product of two nonunits, while a prime element must divide one factor whenever it divides a product. These notions agree in a unique factorization domain but can differ in a general integer ring. Passing to principal ideals sends an element product to an ideal product. Prime ideals in that product need not come from individual prime elements, so unique ideal factorization does not itself give unique element factorization.

In Z[sqrt(−5)], which is the full integer ring of its field, the element six factors as 2 times 3 and also as (1+sqrt(−5))(1−sqrt(−5)). The norm a²+5b² rules out elements of norm two or three, which verifies irreducibility of the factors by multiplicativity. The two factorizations are not related by multiplication by units and reordering. Prime-ideal factorization refines both products into the same collection, with the missing nonprincipal ideal factors carrying the obstruction.

For a nonzero integral ideal I in a number field, its norm is the finite index of I in the integer ring, or equivalently the number of elements in the quotient. Ideal norm is multiplicative. For a principal ideal generated by a nonzero integer-ring element, it equals the absolute field norm of that element. A quotient by a prime ideal is a finite field, but an arbitrary finite quotient ring can have zero divisors and nilpotents. Cardinality alone does not determine its ring structure.

To verify an ideal equality, first prove containment using generators, then compare finite indices or norms. This avoids trusting a formal product of names without checking what the ideals contain. A nonmaximal order need not be integrally closed, and its ideals need not satisfy the full Dedekind factorization and invertibility conclusions. Before applying the theorem, verify the ambient ring rather than assuming that every ring generated by one algebraic integer inherits all properties of the full integer ring.