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

Abstract

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 ) → ⊕ q ∈ P r q Z / Z → 0 where r q ∈ 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.