Cohomological Kernels of Elementary Abelian Degree $p^2$ Extensions

Let $p$ be an odd prime, $F$ a field with a primitive $p^2$th root of unity, and $E=F(\sqrt[p]{b_1},\sqrt[p]{b_2})$ an elementary abelian extension of degree $p^2$. This paper studies the cohomological kernel $H^n(E\mathrm{/}F,\mathbb{Z}\mathrm{/}p\mathbb{Z}):=\mathrm{k}\mathrm{e}\mathrm{r}(H^n(F,\mathbb{Z}\mathrm{/}p\mathbb{Z})\to H^n(E,\mathbb{Z}\mathrm{/}p\mathbb{Z}))$ for all $n$. When $p=3$, using tools of Positselski, a six-term exact sequence is given that is analogous to the $p=2$ case. As an application the quotient $H^n(E\mathrm{/}F,\mathbb{Z}\mathrm{/}3\mathbb{Z})\mathrm{/}{\mathrm{D}\mathrm{e}\mathrm{c}}^n(E\mathrm{/}F,\mathbb{Z}\mathrm{/}3\mathbb{Z})$ where ${\mathrm{D}\mathrm{e}\mathrm{c}}^n(E\mathrm{/}F,\mathbb{Z}\mathrm{/}3\mathbb{Z})$ is the "expected kernel’’ is described. This quotient group is of interest because computations of Tignol [T] that show when $n=2$ nontrivial elements give rise to indecomposible division algebras of exponent $3$ and index $9$.