Алгебралық сандар теориясындағы айырма идеалдары

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. It was introduced by Richard Dedekind in 1882.

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

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 + eνP(e) − 1. The differential exponent can be computed from the orders of the higher ramification groups for Galois extensions:

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).