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
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