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)
“Mozart has the classic purity of light and the blue ocean; Beethoven the romantic grandeur which belongs to the storms of air and sea, and while the soul of Mozart seems to dwell on the ethereal peaks of Olympus, that of Beethoven climbs shuddering the storm-beaten sides of a Sinai. Blessed be they both! Each represents a moment of the ideal life, each does us good. Our love is due to both.”
—Henri-Frédéric Amiel (1821–1881)
“You must drop all your democracy. You must not believe in “the people.” One class is no better than another. It must be a case of Wisdom, or Truth. Let the working classes be working classes. That is the truth. There must be an aristocracy of people who have wisdom, and there must be a Ruler: a Kaiser: no Presidents and democracies.”
—D.H. (David Herbert)
“screenwriter
Policemen so cherish their status as keepers of the peace and protectors of the public that they have occasionally been known to beat to death those citizens or groups who question that status.”
—David Mamet (b. 1947)