# Primes in Kummer rings

1. Nov 10, 2007

### 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.

Last edited: Nov 10, 2007