素指数$p$ -代数的符号长度
Symbol Length of $p$-Algebras of Prime Exponent
摘要：
本文证明了：如果域$F$上的素数阶各向异性齐次多项式形式$p$的最大维数是大于1的有限整数$d$，则$F$上的指数$p$的$p$ -代数的符号长度由$\left \lceil \frac{d-1}{p} \right \rceil-1$上有界，证明了长度为$k$和$\ell$与$(k+\ell) p \geq d-1$的符号代数的每两个张量积都可以修改，使得它们共享一个公共槽.对于$p=2$，我们得到了符号长度的$\frac{u(F)}{2}-1$的上界，当$I_q^3 F=0$。
英文摘要：
We prove that if the maximal dimension of an anisotropic homogeneous polynomial form of prime degree $p$ over a field $F$ with $\operatorname{char}(F)=p$ is a finite integer $d$ greater than 1 then the symbol length of $p$-algebras of exponent $p$ over $F$ is bounded from above by $\left \lceil \frac{d-1}{p} \right \rceil-1$, and show that every two tensor products of symbol algebras of lengths $k$ and $\ell$ with $(k+\ell) p \geq d-1$ can be modified so that they share a common slot. For $p=2$, we obtain an upper bound of $\frac{u(F)}{2}-1$ for the symbol length, which is sharp when $I_q^3 F=0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
