Examples of Ideal Class Groups
- The rings Z, Z, and Z, where ω is a cube root of 1 and i is a fourth root of 1 (i.e. a square root of −1), are all principal ideal domains, and so have class number 1: that is, they have trivial ideal class groups.
- If k is a field, then the polynomial ring k is an integral domain. It has a countably infinite set of ideal classes.
Read more about this topic: Ideal Class Group
Famous quotes containing the words examples of, examples, ideal, class and/or groups:
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“The archetype of all humans, their ideal image, is the computer, once it has liberated itself from its creator, man. The computer is the essence of the human being. In the computer, man reaches his completion.”
—Friedrich Dürrenmatt (1921–1990)
“... in the fierce competition of modern society the only class left in the country possessing leisure is that of women supported in easy circumstances by husband or father, and it is to this class we must look for the maintenance of cultivated and refined tastes, for that value and pursuit of knowledge and of art for their own sakes which can alone save society from degenerating into a huge machine for making money, and gratifying the love of sensual luxury.”
—Mrs. H. O. Ward (1824–1899)
“Trees appeared in groups and singly, revolving coolly and blandly, displaying the latest fashions. The blue dampness of a ravine. A memory of love, disguised as a meadow. Wispy clouds—the greyhounds of heaven.”
—Vladimir Nabokov (1899–1977)