Editing Ideal class group (section)

== History and origin of the ideal class group ==

Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an [[Ideal (ring theory)|ideal]] was formulated. These groups appeared in the theory of [[quadratic form]]s: in the case of [[binary quadratic form|binary]] integral quadratic forms, as put into something like a final form by [[Carl Friedrich Gauss]], a composition law was defined on certain [[equivalence class]]es of forms. This gave a finite [[abelian group]], as was recognised at the time.

Later [[Ernst Kummer]] was working towards a theory of [[cyclotomic field]]s. It had been realised (probably by several people) that failure to complete [[mathematical proof|proofs]] in the general case of [[Fermat's Last Theorem]] by factorisation using the [[roots of unity]] was for a very good reason: a failure of unique factorization – i.e., the [[fundamental theorem of arithmetic]] – to hold in the [[ring (mathematics)|ring]]s generated by those roots of unity was a major obstacle. Out of Kummer's work for the first time came a study of the obstruction to the factorization. We now recognise this as part of the ideal class group: in fact Kummer had isolated the <math>p</math>-[[torsion subgroup|torsion]] in that group for the [[field (mathematics)|field]] of <math>p</math>-roots of unity, for any [[prime number]] <math>p</math>, as the reason for the failure of the standard method of attack on the Fermat problem (see [[regular prime]]).

Somewhat later again [[Richard Dedekind]] formulated the concept of an ideal, Kummer having worked in a different way. At this point the existing examples could be unified. It was shown that while rings of [[algebraic integer]]s do not always have unique factorization into primes (because they need not be [[principal ideal domain]]s), they do have the property that every [[proper ideal]] admits a unique factorization as a product of [[prime ideal]]s (that is, every ring of algebraic integers is a [[Dedekind domain]]). The size of the ideal class group can be considered as a measure for the deviation of a ring from being a principal ideal domain; a ring is a principal ideal domain if and only if it has a trivial ideal class group.