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

When is (q) a prime ideal in [tex]Z(\rho)[/tex] (the Kummer ring) where [tex]\rho = e^{2\pi i /p}[/tex], 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 [tex]x^{p-1} + x^{p-2} + x^{p-3} + ... + 1[/tex] 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.

When is (q) a prime ideal in [tex]Z(\rho)[/tex] (the Kummer ring) where [tex]\rho = e^{2\pi i /p}[/tex], 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 [tex]x^{p-1} + x^{p-2} + x^{p-3} + ... + 1[/tex] 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.

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