Elements need not factor uniquely in rings of algebraic integers, but ideals do. Dedekind domains restore arithmetic by replacing a number with the collection of divisibility conditions it imposes.
Fractional ideals
Definition. For a domain \(R\) with fraction field \(K\), a fractional ideal is a nonzero \(R\)-submodule \(I\subset K\) for which \(aI\subset R\) for some nonzero \(a\in R\). Products are generated by pairwise products: \[ IJ=\left\{\sum_{\ell} x_{\ell}y_{\ell}:x_{\ell}\in I,\ y_{\ell}\in J\right\}. \]
Definition. A Dedekind domain is a Noetherian, integrally closed domain of Krull dimension one; equivalently, every nonzero prime ideal is maximal.
Theorem. The ring of integers \(\mathcal O_K\) of every number field is a Dedekind domain.
Noetherianity follows from its finite \(\mathbb Z\)-module structure, integral closedness from the definition of all algebraic integers in \(K\), and dimension one from the norm: every nonzero quotient \(\mathcal O_K/\mathfrak a\) is finite.
Invertibility and unique factorization
Theorem. Every nonzero fractional ideal in a Dedekind domain is invertible, and every nonzero integral ideal has a unique factorization \[ \mathfrak a=\prod_{\mathfrak p}\mathfrak p^{v_{\mathfrak p}(\mathfrak a)} \] into nonzero prime ideals.
Proof outline. For \(I\ne0\), define \[ I^{-1}=\{x\in K:xI\subset R\}. \] Localization at a nonzero prime \(\mathfrak p\) turns \(R_\mathfrak p\) into a discrete valuation ring, where every ideal is a power of \(\mathfrak pR_\mathfrak p\). Hence \((II^{-1})_\mathfrak p=R_\mathfrak p\) for every \(\mathfrak p\), which implies \(II^{-1}=R\). The local exponents \(v_\mathfrak p(I)\) are zero except finitely often and determine \(I\); this proves existence and uniqueness of the global factorization.
Ideal norm and prime decomposition
Definition. For a nonzero integral ideal \(\mathfrak a\subset\mathcal O_K\), \[ N\mathfrak a=|\mathcal O_K/\mathfrak a|. \]
Proposition. \[ N(\mathfrak a\mathfrak b)=N\mathfrak a\,N\mathfrak b. \] If a rational prime factors as \[ p\mathcal O_K=\mathfrak p_1^{e_1}\cdots\mathfrak p_g^{e_g}, \qquad N\mathfrak p_i=p^{f_i}, \] then \[ \sum_{i=1}^ge_if_i=[K:\mathbb Q]. \]
The integers \(e_i,f_i\) are the ramification indices and residue degrees. They separate how \(p\) branches from how its residue field grows.
Example (Gaussian integers). In \(\mathbb Z[i]\): \[ 2=-i(1+i)^2 \] ramifies; \(p\equiv1\pmod4\) splits into conjugate Gaussian primes; and \(p\equiv3\pmod4\) remains prime. This is equivalent to the solvability of \(x^2\equiv-1\pmod p\).
Dedekind factorization criterion
When \(\mathcal O_K=\mathbb Z[\alpha]\) and \(\overline f=\prod_i\overline g_i^{e_i}\) is the reduction modulo \(p\) of the minimal polynomial of \(\alpha\), under the usual index condition, \[ p\mathcal O_K=\prod_i(p,g_i(\alpha))^{e_i}. \] Polynomial factorization modulo \(p\) therefore predicts ideal factorization in the number field.