From: nikl@mathematik.tu-muenchen.de (Gerhard Niklasch) Newsgroups: sci.math Subject: Re: Unique Factorization Domain Question Date: 11 Nov 1997 12:24:58 GMT In article <3467B2D0.1671@bellsouth.net>, Harden writes: |> Let w_n be a primitive nth root of unity. For all positive integer n>2, |> is Z(w_n) a unique factorization domain? I know it is for n=3, 4, and 6. Far from it. (Note that Z[w_3] is the same as Z[w_6]; we can assume from the start that n is either odd or divisible by 4.) Then Z[w_n] is a UFD for n in {1,3,4,5,7,8,9,11,12,13,15,16,17,19,20,21, 24,25,27,28,32,33,35,36,40,44,45,48,60,84} and for no other values of n. This is a result by Masley from the 1970s, extending earlier work by Montgomery and Uchida, and using Odlyzko's discriminant bounds. For large n, Z[w_n] gets arbitrarily far from being a UFD (in the sense that the class group of fractional ideals modulo principal ideals becomes arbitrarily large). For more details, see Lawrence C Washington, Introduction to Cyclotomic Fields, Springer Graduate Text in Mathematics 83 (1st ed. 1982, 2nd ed. 1996?), Chapter XI. Enjoy, Gerhard -- * Gerhard Niklasch *** Some or all of the con- * http://hasse.mathematik.tu-muenchen.de/~nikl/ ******* tents of the above mes- * sage may, in certain countries, be legally considered unsuitable for consump- * tion by children under the age of 18. Me transmitte sursum, Caledoni... :^/