TITLE: Fermat's last theorem and $\mathbb{Z}[\xi]$
QUESTION [10 upvotes]: 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.

REPLY [7 votes]: 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.