Ideal Number

[Kummer]

For what values does \mathbb{Z}[\zeta] have unique factorization?

I know Kummer shown that \zeta being a 23-rd root of unity fails to have unique factorization.

[robert Ihnot]

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.
http://www.math.niu.edu/~rusin/known-math/97/UFDs

[Kummer]

Does anybody have Introduction to Cyclotomic Extension by Lawrence Washington, I want to see if this is actually true. The site does not look completely reliable. I searched on it on Wikipedia and did not find anything and also on MathWorld.

[robert Ihnot]

You can buy Introduction to Cyclotomic Fields new or used from Amazon.com., and you can compair prices on Yahoo shopping.

[mathwonk]

if w_n means a primitive nth root of 1, i would think n=2 is ok.

[robert Ihnot]

Mathwonk: if w_n means a primitive nth root of 1, i would think n=2 is ok.

Now that that has been brought up, I wondered about it also. Trying to look the link given above over very carefully, I gather that w_n is just w subscript n, where n represents the power and w represents the primitative root.

Writer goes on to say that w_3 is the same as w_6, and omits 6 in his list.* Thus multiplication by units +1 and -1 does not count, which is usually the case in factorization. So then the conclusion I gather is that cases such as N=2,6,14 are omitted because they were, to the author, previously eliminated because they do not represent anything new. (The sum of the roots of X^N-1 =0 is itself 0 and so -1 is already present in the smaller ring.)This is consistant with other writers who say N=23 is the first case of failure.

* (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.)

[CRGreathouse]

Sloane has A005848 (http://www.research.att.com/~njas/sequences/A005848) (,fini,full,nonn,) as 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.