Dedekind domain

Ideal class group

Let be a Dedekind domain, e.g. the ring of integers of some number field, its group of fractional ideals, and be the subgroup of principal ideals. The ideal class group is the quotient group ring

Equivalently, let be the set of nonzero ideals in , and define an equivalence relation on so that for

Then .¹

Proof

First we verify that defines an equivalence relation on , where the only condition that isn’t immediately obvious is ^E3. Suppose so that

Then , so , as required.

Next we show that defines a congruence relation on the monoid . Now suppose such that and . Then so . The quotient monoid is thence well-defined.

Now we show that is in fact a group. Let and . Then so for some ideal .

Finally we show that these groups are isomorphic, letting and denote the constructions with and without fraction ideals respectively. An arbitrary element in is for some fractional ideal . But for some and , so . Therefore we can always use an integral ideal as a representative for an element of , which itself represents an element of . Clearly : hence we have an isomorphism.

Results

• Ideal class group of a number field

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Footnotes

1. 2022. Algebraic number theory course notes, pp. 21–22 ↩