ABSTRACT
The main goal is to prove that all cyclotomic polynomials Φn(x) are irreducible in Q[x], and to see what happens to Φn(x) over Fp when p|n. The irreducibility over Q allows us to conclude that the automorphism group of Q(ζn) over Q (with ζn a primitive nth root of unity) is
Aut(Q(ζn)/Q) ≈ (Z/n)×
by the map (ζn −→ ζan)←− a
The case of prime-power cyclotomic polynomials in Q[x] needs only Eisenstein’s criterion, but the case of general n seems to admit no comparably simple argument. The proof given here uses ideas already in hand, but also an unexpected trick. We will give a different, less elementary, but possibly more natural argument later using p-adic numbers and Dirichlet’s theorem on primes in an arithmetic progression.
