in which the dual of OL is a fractional ideal of L containing OL. The (relative) different δL/K is the inverse of this fractional ideal: it is an ideal of OL.
The relative different encodes the ramification data of the field extension L/K. A prime ideal p of K ramifies in L if and only if it divides the relative discriminant ΔL/K. If
- p = P1e(1) ... Pke(k)
is the factorisation of p into prime ideals of L then Pi divides the relative different δL/K if and only if Pi 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+νP(e)-1.
The different may be defined for an extension of local fields L/K. In this case we may take the extension to be simple, generated by a primitive element α which also generates a power integral basis. If f is the minimal polynomial for α then the different is generated by f'(α).