Class Field Theory

(a) If p ramifies in Q(ζm), then p | m; the ramification index ep of the primes over p is φ(p) = (p− 1)pk−1 if p ‖ m. (b) The prime p is wildly ramified in Q(ζm) if and only if p | m. (c) If p m, then p is unramified. In the ring of integers Z[ζm] of Q(ζm), we obtain pOL = ∏gp i=1 pi. The primes pi have residue class degree fp = [Fp(ζm) : Fp], which (by looking at the Frobenius x 7→ x acting on Fp(ζm) = kpi) is equal to the order of p mod m in (Z/mZ)∗. The number of such primes is gp = #{pi|p} = [(Z/mZ)∗ : 〈p mod m〉]. In particular, a prime splits competely if and only if p ≡ 1 (mod m). If Q ⊂ L ⊂ L = Q(ζm) is a subfield given by a subgroup H ⊂ (Z/mZ)∗, then one has the Artin map