Dedekind Domains and Factorization - Core Definitions

Dedekind domains provide the algebraic framework for recovering unique factorization in rings of integers, where element factorization may fail.

DefinitionDedekind Domain

An integral domain $R$ is a Dedekind domain if it satisfies three conditions:

Noetherian: Every ideal of $R$ is finitely generated
Integrally closed: If $x$ in the field of fractions $K$ satisfies a monic polynomial over $R$ , then $x \in R$
Dimension one: Every nonzero prime ideal of $R$ is maximal

The ring of integers $\mathcal{O}_K$ of any number field $K$ is a Dedekind domain.

ExampleExamples and Non-Examples

Dedekind domains:

$\mathbb{Z}$ (the prototypical example)
$\mathcal{O}_K$ for any number field $K$
$\mathbb{C}[x]$ (polynomial ring in one variable)
Localization $\mathbb{Z}_{(p)}$ at a prime $p$

Not Dedekind domains:

$\mathbb{Z}[\sqrt{-5}]$ is not integrally closed in $\mathbb{Q}(\sqrt{-5})$ , but the integral closure $\mathcal{O}_{\mathbb{Q}(\sqrt{-5})} = \mathbb{Z}\left[\frac{1+\sqrt{-5}}{2}\right]$ is Dedekind
$k[x,y]$ (two variables) fails dimension one: $(x)$ is prime but not maximal

DefinitionFractional Ideals

Let $R$ be a Dedekind domain with field of fractions $K$ . A fractional ideal is a finitely generated $R$ -submodule $I \subseteq K$ such that $dI \subseteq R$ for some nonzero $d \in R$ .

The set of fractional ideals forms a group under multiplication: $I \cdot J = \left\{\sum_{i=1}^n a_i b_i : a_i \in I, b_i \in J\right\}$

The inverse of a fractional ideal $I$ is $I^{-1} = \{x \in K : xI \subseteq R\}$ .

In a Dedekind domain, every nonzero fractional ideal $I$ has an inverse $I^{-1}$ such that $I \cdot I^{-1} = R$ . This group structure is fundamental: the failure of unique factorization for elements is precisely captured by the structure of the ideal class group.

DefinitionPrime Ideal Factorization

In a Dedekind domain $R$ , every nonzero ideal $I$ factors uniquely as a product of prime ideals: $I = \mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_r^{e_r}$ where $\mathfrak{p}_i$ are distinct prime ideals and $e_i \geq 1$ .

This factorization is unique up to reordering. The exponents $e_i$ are called the multiplicities of the primes in $I$ .

Remark

The passage from element factorization to ideal factorization is the key insight of Kummer and Dedekind. While elements like $6 = 2 \cdot 3 = (1+\sqrt{-5})(1-\sqrt{-5})$ may have non-unique factorizations in $\mathbb{Z}[\sqrt{-5}]$ , the corresponding ideals factor uniquely in $\mathcal{O}_{\mathbb{Q}(\sqrt{-5})}$ .

ExampleIdeal Factorization

In $\mathcal{O}_{\mathbb{Q}(i)} = \mathbb{Z}[i]$ , consider the principal ideal $(5)$ . Since $5 = (2+i)(2-i)$ and both factors are irreducible, we have $(5) = (2+i)(2-i) = \mathfrak{p}\bar{\mathfrak{p}}$ where $\mathfrak{p} = (2+i)$ and $\bar{\mathfrak{p}} = (2-i)$ are prime ideals of norm 5.