TITLE: Fermat's Last Theorem and Kummer's Objection
QUESTION [40 upvotes]: In 1847 Lamé had announced that he had proven Fermat's Last Theorem.  This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$.  However, Kummer, proved that when $p=23$ we do not have unique factorization, and in fact Kummer proved this 3 years earlier in 1844.
My question is:  how can you prove that $\mathbb{Z}[\zeta_p]$ does not have unique factorization when $p = 23$, but does for $p < 23$?

REPLY [12 votes]: We know that $\Bbb{Z}[e^{2\pi i/23}]$ is the ring of integers of the 23rd cyclotomic field. Since the ring of integers is a PID iff it is a UFD, it suffices to understand why this ring is not  a PID. For this, you can see my answer given here which proves that the ideal $Q = (2, \frac{1 + \sqrt{-23}}{2})$ is not principal in $\Bbb{Z}[e^{2\pi i/23}]$.