Background: I have stopped doing algebra a long time ago and I am not that interested in the nitty-gritty details of proofs, but I am interested in maths history.
http://mathworld.wolfram.com/CyclotomicInteger.html claims unique factorisation fails for $p>23$
In Kummer's original work, let $\alpha$ be a $p$-th primitive root of unity. $f(\alpha)=a_0+a_1\alpha+\cdots+a_{p-1}\alpha^{p-1}$ and $Nf(\alpha) = f(\alpha)f(\alpha^2)\cdots f(\alpha^{p-1})$, where $N$ is his generalisation of norm from norm of Gaussian integers. This is always a rational integer for $f(\alpha)$.
Kummer defined $f(\alpha)|g(\alpha)$ if $Nf(\alpha)|Ng(\alpha)$
Here is my question:
For $p<23$, it seems cyclotomic integers just work like integers. There is a nice notion of divisibility and prime. In the case $p=23$, using the notion of norm defined by Kummer, there is no factorisation for the number 47. It's norm is $47^{22}$ a not a prime integer, but there is no element in the 23rd-cyclotomic integer with norm 47, so this has no divisor. Is this related to the fact given in the link? (It seems not because the link says $p>23$ and this is a fact for $p=23$, but is there something fundamental I am missing?)
As you correctly guess, unique factorization fails for $p = 23$.
In fact it fails in the ring $\mathbb Z[(1 + \sqrt{-23})/2],$ which is a quadratic subring of the ring of cyclotomic integers for $p = 23$, and although it is not obvious, this implies that it fails in the cyclotomic integer ring. Also, the reason that $47$ witnesses the failure is related to the fact that $47 \equiv 1 \bmod 23$.