Ideal class group

The ideal class group of a Dedekind domain is the group of nonzero fractional ideals modulo principal fractional ideals. Its order measures how far the domain is from having every ideal principal.

Fractional ideal: A nonzero submodule of a ring’s fraction field that becomes an integral ideal after multiplication by some nonzero ring element. These are the group elements before principal ideals are identified.
Unique factorization of ideals: The theorem that every nonzero ideal in a Dedekind domain factors uniquely into prime ideals. It makes prime-ideal exponents available for describing ideal classes.
Class number: The number of elements in a finite ideal class group. It quantifies the total number of distinct ideal classes.
Ernst Kummer: A nineteenth-century German mathematician who developed ideal numbers to address failures of unique factorization. His ideal numbers anticipated the ideal-theoretic framework behind class groups.
Picard group: The group of isomorphism classes of line bundles on a space, with tensor product as its operation. For a Dedekind domain, its invertible-ideal Picard group agrees with the ideal class group.
Principal ideal: An ideal generated by a single ring element. Principal ideals form the subgroup whose quotient defines the class group.
Prime ideal: A proper ideal whose quotient ring is an integral domain. Prime ideals are the irreducible factors in Dedekind-domain ideal factorization.
Minkowski bound: A bound ensuring each ideal class of a number field contains an integral ideal of bounded norm. It reduces class-group computation to finitely many prime ideals.
Richard Dedekind: A German mathematician who developed the modern theory of ideals in algebraic number fields. His ideals replaced Kummer’s ideal numbers with objects inside rings.
Divisor class group: The group of Weil divisors modulo principal divisors on a suitable integral scheme. It generalizes ideal classes from number rings to broader algebraic varieties.

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