Cohomological Kernels for Cyclic by Cyclic Semi-Direct Product Extensions

Let $F$ be a field and $E$ an extension of $F$ with $[E:F]=d$ where the characteristic of $F$ is zero or prime to $d$. We assume $\mu_{d^2}\subset F$ where $\mu_{d^2}$ are the $d^2$th roots of unity. This paper studies the problem of determining the cohomological kernel $H^n(E/F):=\ker(H^n(F,\mu_d) \rightarrow H^n(E,\mu_d))$ (Galois cohomology with coefficients in the $d$th roots of unity) when the Galois closure of $E$ is a semi-direct product of cyclic groups. The main result is a six-term exact sequence determining the kernel as the middle map and is based on tools of Positelski. When $n=2$ this kernel is the relative Brauer group ${\rm Br}(E/F)$, the classes of central simple algebras in the Brauer group of $F$ split in the field $E$. The work of Aravire and Jacob which calculated the groups $H^n_{p^m}(E/F)$ in the case of semidirect products of cyclic groups in characteristic $p$, provides motivation for this work.