Field of Iterated Laurent Series and its Brauer Group

The symbol length of ${}_{p}Br(k(\text{\hspace{0.17em}⁣}({\alpha }_1)\text{\hspace{0.17em}⁣})\dots (\text{\hspace{0.17em}⁣}({\alpha }_n)\text{\hspace{0.17em}⁣}))$ for an algebraically closed field $k$ of $\mathrm{char}(k)\mathrm{\ne }p$ is known to be $\lfloor \frac{n}{2}\rfloor$. We prove that the symbol length for the case of $\mathrm{char}(k)=p$ is rather $n-1$. We also show that pairs of anisotropic quadratic or bilinear $n$-fold Pfister forms over this field need not share an $(n-1)$-fold factor.