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

TL;DR

This paper investigates the bounds on symbol length in higher Galois cohomology groups of fields with characteristic p, revealing new upper limits based on exponent divisibility and specific cases like p=2.

Contribution

It establishes new bounds on symbol length in $H_{p^m}^{n+1}(F)$ for symbols of lower exponent, extending previous results and providing explicit bounds for special cases.

Findings

Bound of $p^n$ for symbol length in $H_{p^{m-1}}^{n+1}(F)$ when exponent divides $p^{m-1}$.

For $n=2$, symbol length is at most $p^r+r-1$ under certain conditions.

When $p=2$, sum of two symbols with exponent dividing $2^{m-1}$ has symbol length at most $(2n+1)2^n$.

Abstract

Let $F$ be a field of characteristic $p>0$. We prove that if a symbol $A=\omega \otimes \beta_1 \otimes \dots \otimes \beta_n$ 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 $A= \omega_1\otimes \beta_1+\cdots+\omega_r\otimes \beta_r$ 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''.