A **Dedekind domain** $D$ is a integral domain which is not a field such that

- $D$ is Noetherian
- every non-zero prime ideal is maximal
- $D$ is integrally closed

The ring of integers for a local number field or a global number field is always a Dedekind domain.

In a Dedekind domain, every non-zero ideal $I$ can be written as a product of non-zero prime ideals, $$I=P_1P_2\cdots P_k,$$ and the product is unique up to the order of the factors. Repeated factors are often grouped, so we write $I=Q_1^{e_1}\cdots Q_g^{e_g}$ where the $Q_i$ are non-zero prime ideals of $D$.

In addition, ever fractional ideal $I$ is invertible in the sense that there exists a fractional ideal $J$ such that $IJ=D$.

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**Knowl status:**

- Review status: beta
- Last edited by John Jones on 2018-08-07 18:52:43

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