On Algorithmic Aspects of the ℓ-Torsion of Brauer Groups

Abstract

We study the algorithmic aspects of the ℓ-torsion of the Brauer group Br(K) of a eld K which contains the ℓ-th roots of unity μℓ, only provided with basic operations on the elements of K. We use a primitive root of unity and nonzero elements in K to represent elements of Br(K)[ℓ]. An algorithm is given to compute the invariant homomorphism on Br(Q&_#8467;(μℓ))[ℓ] for an odd prime ℓ. We also give a remark on potential application of Brauer groups to solve the discrete logarithm problem of multiplicative groups of nite fields.

Quaestiones Mathematicae 35(2012), 297-312