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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.〔, p. 102〕 ==Definition== If ''O''''K'' 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 ''O''''K''. Its discriminant as quadratic form need not be +1 (in fact this happens only for the case ''K'' = Q). Define the ''inverse different'' or ''codifferent'' or ''Dedekind's complementary module'' as the set ''I'' of ''x'' ∈ ''K'' such that tr(''xy'') is an integer for all ''y'' in ''O''''K'', then ''I'' is a fractional ideal of ''K'' containing ''O''''K''. By definition, the different ideal δ''K'' is the inverse fractional ideal ''I''−1: it is an ideal of ''O''''K''. The ideal norm of ''δ''''K'' is equal to the ideal of ''Z'' generated by the field discriminant ''D''''K'' of ''K''. The ''different of an element'' α of ''K'' with minimal polynomial ''f'' is defined to be δ(α) = ''f''′(α) if α generates the field ''K'' (and zero otherwise):〔 we may write : where the α(''i'') run over all the roots of the characteristic polynomial of α other than α itself. The different ideal is generated by the differents of all integers α in ''O''''K''. This is Dedekind's original definition. The different is also defined for an finite degree extension of local fields. It plays a basic role in Pontryagin duality for p-adic fields. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Different ideal」の詳細全文を読む スポンサード リンク
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