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 $H^n(E/F,{\mathbb Z}/p{\mathbb Z}):={\rm ker}(H^n(F,{\mathbb Z}/p{\mathbb Z})\rightarrow H^n(E,{\mathbb Z}/p{\mathbb Z}))$ 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 $H^n(E/F,{\mathbb Z}/3{\mathbb Z})/{\rm Dec}^n(E/F,{\mathbb Z}/3{\mathbb Z})$ where ${\rm Dec}^n(E/F,{\mathbb Z}/3{\mathbb Z})$ 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 $9$.