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In mathematics, a norm variety is a particular type of algebraic variety ''V'' over a field ''F'', introduced for the purposes of algebraic K-theory by Voevodsky. The idea is to relate Milnor K-theory of ''F'' to geometric objects ''V'', having function fields ''F''(''V'') that 'split' given 'symbols' (elements of Milnor K-groups). The formulation is that ''p'' is a given prime number, different from the characteristic of ''F'', and a symbol is the class mod ''p'' of an element : of the ''n''-th Milnor K-group. A field extension is said to ''split'' the symbol, if its image in the K-group for that field is 0. The conditions on a norm variety ''V'' are that ''V'' is irreducible and a non-singular complete variety. Further it should have dimension ''d'' equal to : The key condition is in terms of the ''d''-th Newton polynomial ''s''''d'', evaluated on the (algebraic) total Chern class of the tangent bundle of ''V''. This number : should not be divisible by ''p''2, it being known it is divisible by ''p''. ==Examples== These include (''n'' = 2) cases of the Severi–Brauer variety and (''p'' = 2) Pfister forms. There is an existence theorem in the general case (paper of Markus Rost cited). 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「Norm variety」の詳細全文を読む スポンサード リンク
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