Primes in Kummer rings

[learningphysics]

This is the last question in Elements of Abstract Algebra by Allan Clark.

When is (q) a prime ideal in $$Z(\rho)$$ (the Kummer ring) where $$\rho = e^{2\pi i /p}$$, where p and q are rational primes.

This seems to be a difficult question to answer in general... since considerable effort goes into answering for p =3 alone in the book.

I think we need $$x^{p-1} + x^{p-2} + x^{p-3} + ... + 1$$ to be irreducible over Zq - ring of integers mod q (they show this result specifically for Z(w) the kummer ring for p=3 which I'm generalizing here for all primes... hope I'm correct)...
is there any more we can say immediately without going into specific cases of q ?

Appreciate any help or hints. Thanks.

 

[fresh_42]

Consider the case in which $\mathbb{Z}[\rho]/q\cdot \mathbb{Z}[\rho]$ is no integral domain, i.e. $(q)$ no prime ideal. Then there have to be elements $a=a_0+a_1\rho+\ldots+a_{p-1}\rho^{p-1}\, , \,b=b_0+b_1\rho+\ldots+b_{p-1}\rho^{p-1}$ such that $a\cdot b \in (q)$, i.e. $q\,|\,a\cdot b$.

All other $(q)$ are then prime ideals.