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In mathematics, the Iwasawa algebra Λ(''G'') of a profinite group ''G'' is a variation of the group ring of ''G'' with ''p''-adic coefficients that take the topology of ''G'' into account. More precisely, Λ(''G'') is the inverse limit of the group rings Z''p''(''G''/''H'') as ''H'' runs through the open normal subgroups of ''G''. Commutative Iwasawa algebras were introduced by in his study of Z''p'' extensions in Iwasawa theory, and non-commutative Iwasawa algebras of compact ''p''-adic analytic groups were introduced by . ==Iwasawa algebra of the ''p''-adic integers== In the special case when the profinite group ''G'' is isomorphic to the additive group of the ring of ''p''-adic integers Z''p'', the Iwasawa algebra Λ(''G'') is isomorphic to the ring of the formal power series Z''p''[[''T'']] in one variable over Z''p''. The isomorphism is given by identifying 1 + ''T'' with a topological generator of ''G''. This ring is a 2-dimensional complete Noetherian regular local ring, and in particular a unique factorization domain. It follows from the Weierstrass preparation theorem for formal power series over a complete local ring that the prime ideals of this ring are as follows: *Height 0: the zero ideal. *Height 1: the ideal (''p''), and the ideals generated by irreducible distinguished polynomials (polynomials with leading coefficient 1 and all other coefficients divisible by ''p''). *Height 2: the maximal ideal (''p'',''T''). 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「Iwasawa algebra」の詳細全文を読む スポンサード リンク
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