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In mathematics, a field ''F'' is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial ''P'' over ''F'' has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a 1936 paper ; and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper . The idea itself is attributed to Lang's advisor Emil Artin. Formally, if ''P'' is a non-constant homogeneous polynomial in variables :''X''1, ..., ''X''''N'', and of degree ''d'' satisfying :''d'' < ''N'' then it has a non-trivial zero over ''F''; that is, for some ''x''''i'' in ''F'', not all 0, we have :''P''(''x''''1'', ..., ''x''''N'') = 0. In geometric language, the hypersurface defined by ''P'', in projective space of degree ''N'' − 2, then has a point over ''F''. ==Examples== *Any algebraically closed field is quasi-algebraically closed. In fact, any homogeneous polynomial in at least two variables over an algebraically closed field has a non-trivial zero.〔Fried & Jarden (2008) p.455〕 *Any finite field is quasi-algebraically closed by the Chevalley–Warning theorem.〔Fried & Jarden (2008) p.456〕〔Serre (1979) p.162〕〔Gille & Szamuley (2006) p.142〕 *Algebraic function fields over algebraically closed fields are quasi-algebraically closed by Tsen's theorem.〔〔Gille & Szamuley (2006) p.143〕 *The maximal unramified extension of a complete field with a discrete valuation and a perfect residue field is quasi-algebraically closed.〔 *A complete field with a discrete valuation and an algebraically closed residue field is quasi-algebraically closed by a result of Lang.〔〔Gille & Szamuley (2006) p.144〕 * A pseudo algebraically closed field of characteristic zero is quasi-algebraically closed.〔Fried & Jarden (2008) p.462〕 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「Quasi-algebraically closed field」の詳細全文を読む スポンサード リンク
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