Prime ideals in $\mathbb Z[\zeta_{18}]$?

Question

I'm trying to write the ideal $(8)$ as a product of prime ideals in $\mathbb Z[\zeta_{18}]$.

So far I have that the minimal polynomial of $\zeta_{18}$ is $x^6-x^3+1$ but I'm confused about how to progress from here...

Any advice?

Answer 1

I was expecting and hoping that someone else would answer your question here. I guess it falls to me.

Your ring is $\Bbb Z[\zeta_9]$, the ring of integers of the ninth cyclotomic field. The only ramified primes here are the divisors of the index, thus $3$ only. So the factorization of $(8)=(2)^3\subset\Bbb Z$ depends only on that of $(2)$, which is now necessarily unramified. The only question is whether $(2)$ splits in $\Bbb Q[\zeta_9]$.

I now claim that at $2$, the residue field extension degree is six; that is, that the field extension $\Bbb F_2[\zeta_9]\supset\Bbb F_2$ has degree six. But the smallest power $2^n$ for which $9|(2^n-1)$ is $64=2^6$. Thus the $f$ in the expression $n=e\,fg$ (product of ramification, res. fld. deg., and number of extensions) gives $f=6$ and thus $g=1$, so we get $(8)=\mathfrak P_2^3$.