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)
“For most Northerners, Texas is the home of real men. The cowboys, the rednecks, the outspoken self-made right-wing millionaires strike us as either the best or worst examples of American manliness.... The ideal is not an illusion nor is it contemptible, no matter what damage it may have done. Many people who scorn it in conversation want to submit to it in bed. Those who believe machismo reeks of violence alone choose to forget it once stood for honor as well.”
—Edmund White (b. 1940)
“The class at Harvard in 1851, have purchased for themselves a notoriety they will not covet in years to come.”
—Harriot K. Hunt (1805–1875)
“In properly organized groups no faith is required; what is required is simply a little trust and even that only for a little while, for the sooner a man begins to verify all he hears the better it is for him.”
—George Gurdjieff (c. 1877–1949)