I'm studying for my exam (tomorrow!) and I came across the following problem that I'm not sure how to approach.
Give an example if possible, and briefly explain why your example works. If no such example exists, briefly explain why this is so.
i) an integer $n \geq 5$ satisfying the property that $\Phi_n(x)$ is irreducible over $F_p$ for all primes p not dividing n;
ii) an integer $n \geq 5$ satisfying the property that $\Phi_n(x)$ is reducible over $F_p$ for all primes p not dividing n.
($F_p$ is the finite field with $p$ elements).
For the first one, I want to say that $5$ itself will work since its prime and so relatively prime to all other primes. For the second one, I want to say that its not possible since no matter what we pick, bigger prime numbers are relatively prime. But, I'm not sure at all about these.
