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Euclid's Algorithm in Cyclotomic Fields
Author(s) -
Lenstra H. W.
Publication year - 1975
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-10.4.457
Subject(s) - citation , mathematics , algebra over a field , library science , information retrieval , computer science , algorithm , combinatorics , pure mathematics
Since Z[Cm] = Z[(2mJ for m odd, this gives eleven non-isomorphic Euclidean rings, corresponding tom= I, 3, 4, 5, 7, 8, 9, 11, 12, 15, 20. The cases m = l, 3, 4, 5, 8, 12 are more or les(classical [2 (pp. l 17-118 and pp. 391--393); 8; 5 (pp. 228-231 ); 3 (chapters 12, 14 and 15); 4; 7]. The other five cases are apparently new. For m even, the ring Z[(m] has class number one if and only if (m) :;:; 20 or m = 70, 84 or 90, see [6]. So there are exactly thirty non-isomorphic rings Z[(m] which admit unique factorization. If certain generalized Riemann hypotheses would hold, then all these thirty rings would be Euclidean for some function different from the norm map [9].