Different Ideal - Ramification

Ramification

The relative different encodes the ramification data of the field extension L / K. A prime ideal p of K ramifies in L if the factorisation of p in L contains a prime of L to a power higher than 1: this occurs if and only if p divides the relative discriminant Δ_{L / K}. More precisely, if

p = P₁e(1) ... P_ke(k)

is the factorisation of p into prime ideals of L then P_i divides the relative different δ_{L / K} if and only if P_i is ramified, that is, if and only if the ramification index e(i) is greater than 1. The precise exponent to which a ramified prime P divides δ is termed the differential exponent of P and is equal to e − 1 if P is tamely ramified: that is, when P does not divide e. In the case when P is wildly ramified the differential exponent lies in the range e to e + ν_P(e) − 1. The differential exponent can be computed from the orders of the higher ramification groups for Galois extensions: