Dedekind Domains and Factorization - Key Properties

Dedekind domains possess remarkable properties that distinguish them from general integral domains and enable ideal-theoretic arithmetic.

DefinitionIdeal Norm

For a nonzero ideal $I$ in $\mathcal{O}_K$ , the norm is defined as $N(I) = |\mathcal{O}_K / I|$ the cardinality of the quotient ring. For principal ideals, $N((\alpha)) = |N_{K/\mathbb{Q}}(\alpha)|$ .

The norm is multiplicative: $N(IJ) = N(I)N(J)$ . For prime ideals $\mathfrak{p}$ lying over a rational prime $p$ , we have $N(\mathfrak{p}) = p^f$ where $f$ is the inertia degree.

ExampleComputing Ideal Norms

In $\mathbb{Z}[i]$ , the ideal $\mathfrak{p} = (2+i)$ has norm $N(\mathfrak{p}) = |\mathbb{Z}[i]/(2+i)| = |N(2+i)| = |4 - (-1)| = 5$

For the ideal $(1-i)$ in $\mathbb{Z}[i]$ , we compute $N((1-i)) = |N(1-i)| = |1 - (-1)| = 2$ Note that $(2) = (1-i)^2$ in $\mathbb{Z}[i]$ , consistent with $N((2)) = 4 = 2^2$ .

DefinitionSplitting of Primes

Let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$ lying above a rational prime $p$ (meaning $\mathfrak{p} \cap \mathbb{Z} = (p)$ ). The factorization of $(p)$ in $\mathcal{O}_K$ is $(p) = \mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_g^{e_g}$

where:

$e_i$ is the ramification index of $\mathfrak{p}_i$
$f_i$ is the inertia degree (residue field degree) of $\mathfrak{p}_i$
These satisfy $\sum_{i=1}^g e_i f_i = n = [K : \mathbb{Q}]$

A prime $p$ splits completely if $e_i = f_i = 1$ for all $i$ (so $g = n$ ). It ramifies if some $e_i > 1$ , and is inert if $g = 1$ and $e_1 = 1, f_1 = n$ .

ExampleSplitting in Quadratic Fields

For $K = \mathbb{Q}(\sqrt{d})$ and prime $p \nmid 2d$ :

$p$ splits: $(p) = \mathfrak{p}\bar{\mathfrak{p}}$ if $\left(\frac{d}{p}\right) = 1$
$p$ is inert: $(p)$ remains prime if $\left(\frac{d}{p}\right) = -1$
$p$ ramifies: $(p) = \mathfrak{p}^2$ if $p | d$

For $K = \mathbb{Q}(i)$ and $p = 5$ : Since $\left(\frac{-1}{5}\right) = 1$ , we have $(5) = (2+i)(2-i)$ .

Remark

The Chinese Remainder Theorem holds in Dedekind domains: if $I_1, \ldots, I_n$ are pairwise coprime ideals (meaning $I_i + I_j = R$ for $i \neq j$ ), then $\mathcal{O}_K / (I_1 \cdots I_n) \cong \mathcal{O}_K/I_1 \times \cdots \times \mathcal{O}_K/I_n$ This isomorphism is crucial for computing with ideals and solving congruences.

DefinitionInvertibility

A fractional ideal $I$ is invertible if there exists a fractional ideal $J$ such that $IJ = R$ . In a Dedekind domain, every nonzero fractional ideal is invertible.

The group of fractional ideals modulo principal ideals is the ideal class group $\text{Cl}(K)$ , measuring the failure of unique factorization.

The class number $h_K = |\text{Cl}(K)|$ is finite for number fields, a deep result combining geometry of numbers and analysis. When $h_K = 1$ , the ring $\mathcal{O}_K$ is a principal ideal domain (PID), hence a unique factorization domain (UFD).