Dedekind domain

A Dedekind domain is an integral domain that has the following three properties: (i) Noetherian, (ii) Integrally closed, (iii) All non-zero prime ideals are maximal.

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Big list of equivalent characterizations:

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“To divide is to contain” for any commutative ring; this is an iff for Dedekind domains. # Theorems/Results

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Exercises

Show that if \(R\) is Dedekind then all of its localizations at maximal ideals are DVRs.
Use the fact that \(R\coloneqq{\mathbf{C}}[t]\) is a Dedekind domain to show that any complex polynomial has only finitely many roots.
Prove that the integral closure of a Dedekind domain in a finite extension of its fraction field is also a Dedekind domain.
Show that if \(K\) is a number field then \({\mathcal{O}}_K\) is a Dedekind domain.

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scheme

Dedekind domain
ramification index

The ramification is tame when the ramification indices \(e_{i}\) are all relatively prime to the residue characteristic \(p\) of \(p\), otherwise wild. A finite generically etale extension \(B / A\) of Dedekind domains is tame if and only if the trace \(\operatorname{Tr}: B \rightarrow A\) is surjective. # Results
order

Any Dedekind domain which is not a field is an order.

The integral closure of an order is always Dedekind.
files without tags

Unsorted/Dedekind domain.md
examples of K theory rings

For \(R\) a Dedekind domain with \(K = \operatorname{ff}(R)\), we have \({\mathsf{K}}_0(R) \cong {\mathbf{Z}}\oplus \operatorname{Cl} (R)\), the class group.
class group

Show that a Dedekind domain \(R\) is a UFD iff \({ \operatorname{cl}} (R) = 1\).