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In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can be infinite. ==The inequality== Let ''A'' = ''K''<''x''1, ..., ''x''''n''> be the free algebra over a field ''K'' in ''n'' = ''d'' + 1 non-commuting variables ''x''''i''. Let ''J'' be the 2-sided ideal of ''A'' generated by homogeneous elements ''f''''j'' of ''A'' of degree ''d''''j'' with :2 ≤ ''d''1 ≤ ''d''2 ≤ ... where ''d''''j'' tends to infinity. Let ''r''''i'' be the number of ''d''''j'' equal to ''i''. Let ''B''=''A''/''J'', a graded algebra. Let ''b''''j'' = dim ''B''''j''. The ''fundamental inequality'' of Golod and Shafarevich states that :: As a consequence: * ''B'' is infinite-dimensional if ''r''''i'' ≤ ''d''2/4 for all ''i'' * if ''B'' is finite-dimensional, then ''r''''i'' > ''d''2/4 for some ''i''. 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「Golod–Shafarevich theorem」の詳細全文を読む スポンサード リンク
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