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In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring ''R'', its representation space is generally denoted by ''R''(1) (that is, it is a representation ). ==''p''-adic cyclotomic character== If ''p'' is a prime, and ''G'' is the absolute Galois group of the rational numbers, the ''p''-adic cyclotomic character is a group homomorphism : where Z''p''× is the group of units of the ring of p-adic integers. This homomorphism is defined as follows. Let ''ζ''''n'' be a primitive ''p''''n'' root of unity. Every ''p''''n'' root of unity is a power of ''ζ''''n'' uniquely defined as an element of the ring of integers modulo ''p''''n''. Primitive roots of unity correspond to the invertible elements, i.e. to (Z/''p''''n'')×. An element ''g'' of the Galois group ''G'' sends ''ζ''''n'' to another primitive ''p''''n'' root of unity : where ''a''''g'',''n'' ∈ (Z/''p''''n'')×. For a given ''g'', as ''n'' varies, the ''a''''g'',''n'' form a comptatible system in the sense that they give an element of the inverse limit of the (Z/''p''''n'')×, which is Zp×. Therefore, the ''p''-adic cyclotomic character sends ''g'' to the system (''a''''g'',''n'')''n'', thus encoding the action of ''g'' on all ''p''-power roots of unity. In fact, is a continuous homomorphism (where the topology on ''G'' is the Krull topology, and that on Z''p''× is the p-adic topology). 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「Cyclotomic character」の詳細全文を読む スポンサード リンク
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