Field of Iterated Laurent Series and its Brauer Group

Abstract

The symbol length of ${}_{p}Br(k((\hat{I}{\pm }_1))\dots ((\hat{I}{\pm }_n)))$ for an algebraically closed field $k$ of $\mathrm{char}(k)\ne p$ 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.