Unique Factorization in $\mathbb{Z}[\zeta]$

[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 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.