On the Dec group of finite Abelian Galois extensions over global fields

If K/F is a finite Abelian Galois extension of global fields whose Galois group has exponent t, we prove that there exists a short exact sequence 0→Dec(K/F)→Brt(K/F)→⊕q∈PrqZ/Z→0 where rq∈Q and P is a finite set of primes of F that is empty if t is square free. In particular, we obtain that if t is square free, then Dec(K/F)=Brt(K/F) which we use to show that prime exponent division algebras over Henselian valued fields with global residue fields are isomorphic to a tensor product of cyclic algebras. Finally, we construct a counterexample to the result for higher exponent division algebras.