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) -> Br-t (K/F) -> circle plus(q epsilon P) r(q)Z/Z -> 0 where r(q) epsilon 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) = Br-t(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. (C) 2009 Elsevier Inc. All rights reserved.