Cohomological Kernels for Cyclic, Dihedral and Other Extensions

Abstract

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 thesis 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 \cite{Positselski}. 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 $E$. In the case where $E$ has degree $d$ and the Galois closure of $E$, \tE has Galois group {\rm Gal}(\tE/F) a dihedral group of degree $2d$, then work of Rowen and Saltman (1982) \cite{RowenSaltman} shows every division algebra $D$ of index $d$ split by $E$ is cyclic over $F$ (that is, $D$ has a cyclic maximal subfield.) This work, along with work of Aravire and Jacob (2008, 2018) \cite{AJ08} \cite{AJ18} which calculated the groups $H^n_{p^m}(E/F)$ in the case of semi-direct products of cyclic groups in characteristic $p$, provides motivation for this work