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TheoremStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31
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A Dedekind domain is a PID exactly when its class group is trivial

Statement

Assume the Axiom of Choice.

Let R be a Dedekind domain. Then R is a principal ideal domain if and only if its ideal class group Cl⁡(R) is trivial.

Facts & Assumptions

Given: The Axiom of Choice and a Dedekind domain R.

[F1]

A principal ideal domain is a domain in which every ideal is principal (Principal ideal domain).

[F2]

The class group is the quotient of nonzero fractional ideals by principal fractional ideals (The ideal class group).

[L1]

Nonzero ideals factor uniquely into prime powers in a Dedekind domain (Unique factorization of nonzero fractional ideals into prime powers).

[L2]

Multiplication descends to the class-group quotient (The ideal class group quotient is well defined).

Proof

technique · direct
1.1

If R is a PID, then every nonzero fractional ideal is principal, so the quotient in [F2] has exactly one class. Thus Cl⁡(R) is trivial.

F1F2given
1.2

Conversely, assume Cl⁡(R) is trivial. Then every nonzero prime ideal has trivial class and is therefore principal. By [L1], every nonzero integral ideal is a finite product of principal prime ideals, hence principal. Therefore [F1] makes R a PID.

F1F2L1L2given
2.1

This proves the equivalence.

step 1.1step 1.2∎

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