Fermat's last theorem and $\mathbb{Z}[\xi]$ I heard that one can prove special cases of FLT by using unique factorization in $\mathbb{Z}[\xi]$ (whenever this is possible), where $\xi$ is a primitive $n$-th root of unity. How can one do this in detail, and is it known for which natural $n$ $\mathbb{Z}[\xi]$ is not a UFD? I mean assuming that $n$ is such that $\mathbb{Z}[\xi]$, how does one proceed?
A reference is good enough.
 A: Since you told that "a reference is good enough", since first chapter of Marcus' book "Number Field" is the very right place to go and since writing down a complete and detailed answer is a huge work -even if rewarded with +$500$ as already pointed out-, I'm writing this reference suggestion as an answer, simply because reasonably this IS a good answer.
A: I think you refer to a theorem of Kummer: Fermat's Last Theorem is true for an odd prime $p$ if and only if $p$ doe not divide the class number of the cyclotomic extension $\mathbf Q(\zeta_p)$, i. e. the order of the group of fractionary ideals of this field modulo principal ideals.
Such a prime number is called a regular prime. 
Kummer criterion:
An odd prime $p$ is regular if and only if $p$ does not divide the numerators of the Bernoulli numbers: $\, B_2, B_4,\dots, B_{p-3}$.
All odd primes up to $31$ regular. It is not known if there is an infinity of regular primes.
In case $\mathbf Z(\zeta_p)$ is a UFD, which is equivalent to being a PID, the class number is equal to $1$, hence  $p$ doen't divide it, so Fermat Last Theorem is true for these primes, of which the complete list is:
$$\{3,5,7,11,13,17,19\}$$
