The Cyclicity of Tensor Products of Cyclic $p$-Algebras
Adam Chapman
TL;DR
The paper advances a constructive treatment of Albert's theorem by providing a computational method to obtain explicit symbol presentations for the cyclic representative of tensor products of cyclic $p$-algebras in characteristic $p>0$, using Witt-vector technology and a map from $W_m \Omega^1_F$ to the $p^m$-torsion of the Brauer group. It proves that the tensor product of a cyclic algebra of degree $p^m$ with a cyclic algebra of degree $p$ is cyclic of degree $p^{m+1}$, giving an explicit formula $[ (\delta,\tau),\delta\gamma^{p^m})_{p^{m+1},F}$. This cyclicity result extends by induction to all tensor products of cyclic $p$-algebras of prime degree, and the general case is handled by a computational, inductive approach that yields a Brauer-equivalent cyclic representative for arbitrary products. The work provides concrete tools to compute Brauer classes in positive characteristic, with potential implications for explicit descriptions of CSA decompositions and Brauer group computations.
Abstract
We revisit the famous theorem of Albert's on the cyclicity of tensor products of cyclic $p$-algebras. In the case of tensor products of cyclic $p$-algebras of prime degree, we provide an explicit computation of the resulting cyclic algebra in symbol algebra terms.