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)
“Every epoch which seeks renewal first projects its ideal into a human form. In order to comprehend its own essence tangibly, the spirit of the time chooses a human being as its prototype and raising this single individual, often one upon whom it has chanced to come, far beyond his measure, the spirit enthuses itself for its own enthusiasm.”
—Stefan Zweig (18811942)
“We are the only class in history that has been left to fight its battles alone, unaided by the ruling powers. White labor and the freed black men had their champions, but where are ours?”
—Elizabeth Cady Stanton (1815–1902)
“If we can learn ... to look at the ways in which various groups appropriate and use the mass-produced art of our culture ... we may well begin to understand that although the ideological power of contemporary cultural forms is enormous, indeed sometimes even frightening, that power is not yet all-pervasive, totally vigilant, or complete.”
—Janice A. Radway (b. 1949)