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 was formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as put into something like a final form by Gauss, a composition law was defined on certain equivalence classes of forms. This gave a finite abelian group, as was recognised at the time.
Later Kummer was working towards a theory of cyclotomic fields. It had been realised (probably by several people) that failure to complete 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 the fundamental theorem of arithmetic to hold in the rings 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 factorisation. We now recognise this as part of the ideal class group: in fact Kummer had isolated the p-torsion in that group for the field of p-roots of unity, for any prime number p, as the reason for the failure of the standard method of attack on the Fermat problem (see regular prime).
Somewhat later again Dedekind formulated the concept of 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 integers do not always have unique factorization into primes (because they need not be principal ideal domains), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals (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 domain; a ring is a principal domain if and only if it has a trivial ideal class group.
Read more about this topic: Ideal Class Group
Famous quotes containing the words history and, history, origin, ideal, class and/or group:
“There is a constant in the average American imagination and taste, for which the past must be preserved and celebrated in full-scale authentic copy; a philosophy of immortality as duplication. It dominates the relation with the self, with the past, not infrequently with the present, always with History and, even, with the European tradition.”
—Umberto Eco (b. 1932)
“This is the greatest week in the history of the world since the Creation, because as a result of what happened in this week, the world is bigger, infinitely.”
—Richard M. Nixon (19131995)
“Art is good when it springs from necessity. This kind of origin is the guarantee of its value; there is no other.”
—Neal Cassady (19261968)
“If we love-and-serve an ideal we reach backward in time to its inception and forward to its consummation. To grow is sometimes to hurt; but who would return to smallness?”
—Sarah Patton Boyle, U.S. civil rights activist and author. The Desegregated Heart, part 3, ch. 3 (1962)
“Alas for the cripple Practice when it seeks to come up with the bird Theory, which flies before it. Try your design on the best school. The scholars are of all ages and temperaments and capacities. It is difficult to class them, some are too young, some are slow, some perverse. Each requires so much consideration, that the morning hope of the teacher, of a day of love and progress, is often closed at evening by despair.”
—Ralph Waldo Emerson (18031882)
“For me, as a beginning novelist, all other living writers form a control group for whom the world is a placebo.”
—Nicholson Baker (b. 1957)