Class field theory summarized

Class Field Theory

Class Field Theory

Computational class field theory

Class field theory furnishes an intrinsic description of the abelian extensions of a number field which is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such extensions.

Class field theory

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...