有理平凡二次空间是局部平凡的：III
Rationally trivial quadratic spaces are locally trivial:III
摘要：
设R是包含域的正则半局部整环，使得所有的剩余域都是无限的.设K是R的分式域。设q是R上自由秩为n的R-模P上的一个二次空间，使得射影二次曲面q=0在R上是光滑的。证明了若K上的二次空间q是迷向的，则在自由秩n的R-模P中存在幺模向量v，使得q（v）=0.如果R的特征为2，则在偶数n的情况下，我们对q的假设等价于q是\cite{Kn}意义下的非奇异空间，而在奇数n > 2的情况下，我们对q的假设等价于q是\cite{Kn}意义下的半正则空间。
英文摘要：
Let R be a regular semi-local domain containing a field such that all the residue fields are infinite. Let K be the fraction field of R. Let q be a quadratic space over R on a free rank n R-module P such that the projective quadric q=0 is smooth over R. It is proved that if the quadratic space q is isotropic over K, then there is a unimodular vector v in the free rank n R-module P such that q(v)=0. If characteristic of R is 2, then in the case of even n our assumption on q is equivalent to the one that q is a non-singular space in the sense of \cite{Kn} and in the case of odd n > 2 our assumption on q is equivalent to the one that q is a semi-regular in the sense of \cite{Kn}.
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