Let F be a field and E an extension of F with [E:F]=d where the
characteristic of F is zero or prime to d. We assume μd2⊂F
where μd2 are the d2th roots of unity. This paper studies the
problem of determining the cohomological kernel Hn(E/F):=ker(Hn(F,μd)→Hn(E,μd)) (Galois cohomology with coefficients in the dth
roots of unity) when the Galois closure of E is a semi-direct product of
cyclic groups. The main result is a six-term exact sequence determining the
kernel as the middle map and is based on tools of Positelski. When n=2 this
kernel is the relative Brauer group Br(E/F), the classes of central
simple algebras in the Brauer group of F split in the field E. The work of
Aravire and Jacob which calculated the groups Hpmn(E/F) in the case of
semidirect products of cyclic groups in characteristic p, provides motivation
for this work