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In mathematics, the Tsen rank of a field describes conditions under which a system of polynomial equations must have a solution in the field. The concept is named for C. C. Tsen, who introduced their study in 1936. We consider a system of ''m'' polynomial equations in ''n'' variables over a field ''F''. Assume that the equations all have constant term zero, so that (0, 0, ... ,0) is a common solution. We say that ''F'' is a T''i''-field if every such system, of degrees ''d''1, ..., ''d''''m'' has a common non-zero solution whenever : The ''Tsen rank'' of ''F'' is the smallest ''i'' such that ''F'' is a T''i''-field. We say that the Tsen rank of ''F'' is infinite if it is not a T''i''-field for any ''i'' (for example, if it is formally real). ==Properties== * A field has Tsen rank zero if and only if it is algebraically closed. * A finite field has Tsen rank 1: this is the Chevalley–Warning theorem. * If ''F'' is algebraically closed then rational function field ''F''(''X'') has Tsen rank 1. * If ''F'' has Tsen rank ''i'', then the rational function field ''F''(''X'') has Tsen rank at most ''i'' + 1. * If ''F'' has Tsen rank ''i'', then an algebraic extension of ''F'' has Tsen rank at most ''i''. * If ''F'' has Tsen rank ''i'', then an extension of ''F'' of transcendence degree ''k'' has Tsen rank at most ''i'' + ''k''. * There exist fields of Tsen rank ''i'' for every integer ''i'' ≥ 0. 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「Tsen rank」の詳細全文を読む スポンサード リンク
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