Classes in ${\mathrm{H}}_{p^m}^{n+1}(F)$ of lower exponent

Abstract

Let $F$ be a field of characteristic $p>0$. We prove that if a symbol $A=\ddot{I}‰\otimes {\hat{I}}_1^2\otimes \cdots \otimes {\hat{I}}_n^2$ in $H_{p^m}^{n+1}(F)$ is of exponent dividing $p^{m-1}$, then its symbol length in $H_{p^{m-1}}^{n+1}(F)$ is at most $p^n$. In the case $n=2$ we also prove that if $A=\ddot{I}{‰}_1\otimes {\hat{I}}_1^2+\cdots +\ddot{I}{‰}_r\otimes {\hat{I}}_r^2$ in $H_{p^m}^2(F)$ satisfies $\exp (A)|p^{m-1}$, then the symbol length of $A$ in $H_{p^{m-1}}^2(F)$ is at most $p^r+r-1$. We conclude by looking at the case $p=2$ and proving that if $A$ is a sum of two symbols in $H_{2^m}^{n+1}(F)$ and $\exp A|2^{m-1}$, then the symbol length of $A$ in $H_{2^{m-1}}^{n+1}(F)$ is at most $(2n+1)2^n$. Our results use norm conditions in characteristic $p$ in the same manner as Matrzi in his paper ``On the symbol length of symbols''.