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Class numbers
This article is under construction.
Please do not rely on any information it contains.
The class number of the ring of algebraic integers of tells how far from or how close that ring is to being a unique factorization domain ( is some algebraic integer).
A class number of 1 means the domain is a unique factorization domain (UFD, not necessarily Euclidean, however). A class number of 2 means the domain is not a UFD, but does guarantee that whenever distinct factorizations a particular number in that domain may have, they each involve as many non-unit irreducible factors. A class number of 3 or higher means the domain is not a UFD and there may be no guarantees of similarities between distinct factorizations of a particular number in that domain.
Class numbers have been studied most thoroughly in the case of , where is a squarefree integer, positive or negative.
A-number | ||
1 | −163, −67, −43, −19, −11, −7, −3, −2, −1, 1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 22, 23, 29, 31, ... | A061574 |
2 | −5, −6, −10, −13, −15, −22, −35, −37, −51, −58, −91, −115, −123, −187, −235, −267, −403, −427 | A005847 |
10, 15, 26, 30, 34, 35, 39, 42, 51, 55, 58, 65, 66, 70, 74, 78, 85, 87, 91, 95, 102, 105, 106, 110, ... | A029702 | |
3 | −23, −31, −59, −83, −107, −139, −211, −283, −307, −331, −379, −499, −547, −643, −883, −907 | A006203 |
79, 142, 223, 229, 254, 257, 321, 326, 359, 443, 469, 473, 659, 733, 761, 839, 934, 993, 1091, ... | A029703 | |
4 | −14, −17, −21, −30, −33, −34, −39, −42, −46, −55, −57, −70, −73, −78, −82, −85, −93, −97, −102, ... (contains only thirty-five more numbers) |
A046085 |
82, 130, 145, 170, 195, 210, 219, 231, 255, 274, 290, 291, 322, 323, 330, 370, 390, 410, 434, 435, ... | A029704 |
Notes: Sequences consisting only of negative numbers are shown multiplied by −1 in their respective entries. Ellipses indicate sequences not known to be finite unless otherwise noted.