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In mathematics, the universal invariant or ''u''-invariant of a field describes the structure of quadratic forms over the field. The universal invariant ''u''(''F'') of a field ''F'' is the largest dimension of an anisotropic quadratic space over ''F'', or ∞ if this does not exist. Since formally real fields have anisotropic quadratic forms (sums of squares) in every dimension, the invariant is only of interest for other fields. An equivalent formulation is that ''u'' is the smallest number such that every form of dimension greater than ''u'' is isotropic, or that every form of dimension at least ''u'' is universal. ==Examples== * For the complex numbers, ''u''(C) = 1. * If ''F'' is quadratically closed then ''u''(''F'') = 1. * The function field of an algebraic curve over an algebraically closed field has ''u'' ≤ 2; this follows from Tsen's theorem that such a field is quasi-algebraically closed.〔Lam (2005) p.376〕 * If ''F'' is a nonreal global or local field, or more generally a linked field, then ''u''(''F'') = 1,2,4 or 8.〔Lam (2005) p.406〕 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「U-invariant」の詳細全文を読む スポンサード リンク
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