In algebraic number theory, the different ideal (sometimes simply the different) is defined to measure the (possible) lack of duality in the ring of integers of an algebraic number field K, with respect to the field trace.
It then encodes the ramification data for prime ideals of the ring of integers.
[1][2] If OK is the ring of integers of K, and tr denotes the field trace from K to the rational number field Q, then is an integral quadratic form on OK. Its discriminant as quadratic form need not be +1 (in fact this happens only for the case K = Q).
[9] The different is also defined for a finite degree extension of local fields.
It plays a basic role in Pontryagin duality for p-adic fields.
[15] 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 ΔL / K. More precisely, if 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.
[16][18][19] The differential exponent can be computed from the orders of the higher ramification groups for Galois extensions:[20]