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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 X1, ..., XN, and of degree d satisfying

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  • Corps quasi-algébriquement clos (fr)
  • Quasi-algebraically closed field (en)
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  • En mathématiques, un corps K est dit quasi-algébriquement clos si tout polynôme homogène P sur K non constant possède un zéro non trivial dès que le nombre de ses variables est strictement supérieur à son degré, autrement dit : si pour tout polynôme P à coefficients dans K, homogène, non constant, en les variables X1, …, XN et de degré d < N, il existe un zéro non trivial de P sur K, c'est-à-dire des éléments x1, …, xN de K non tous nuls tels que P(x1, …, xN) = 0. En termes géométriques, l'hypersurface définie par P, dans l'espace projectif de dimension N – 1, possède alors un point sur K. (fr)
  • 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 X1, ..., XN, and of degree d satisfying (en)
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  • En mathématiques, un corps K est dit quasi-algébriquement clos si tout polynôme homogène P sur K non constant possède un zéro non trivial dès que le nombre de ses variables est strictement supérieur à son degré, autrement dit : si pour tout polynôme P à coefficients dans K, homogène, non constant, en les variables X1, …, XN et de degré d < N, il existe un zéro non trivial de P sur K, c'est-à-dire des éléments x1, …, xN de K non tous nuls tels que P(x1, …, xN) = 0. En termes géométriques, l'hypersurface définie par P, dans l'espace projectif de dimension N – 1, possède alors un point sur K. Cette notion a été d'abord étudiée par Chiungtze Tsen, un étudiant d'Emmy Noether, dans un article de 1936, puis par Serge Lang en 1951 dans sa thèse. L'idée elle-même est attribuée à Emil Artin. (fr)
  • 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 X1, ..., XN, and of degree d satisfying d < N then it has a non-trivial zero over F; that is, for some xi in F, not all 0, we have P(x1, ..., xN) = 0. In geometric language, the hypersurface defined by P, in projective space of degree N − 2, then has a point over F. (en)
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