Show that Φ_m is irreducible

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https://preview.redd.it/pqzuml8uqtp91.png?width=1505&format=png&auto=webp&s=5ee7d8b1c3872916329eae6b8bd5527610d6bc10

Hi everyone, don't understand the solution.

1. How do they deduce that f,g ∈Z[X]? I mean f is monic, then g must also be monic so the content is automatically 1.
2. Don't see why g(x) and f(x) have the same root in overline F_p. Btw not sure why overline F_p actually is. Maybe we've p| g(ς_m),f(ς_m). Therefore they've the same root mod p which occurs twice.

Oh man I've lots problems with this. Hope so. can help me.