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 field K which contains the l-th roots of unity µ _l, 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_l (µ _l))[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 finite fields.