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

TL;DR

This paper investigates the structure of the Dec group in finite abelian Galois extensions over global fields, establishing conditions for its equality with certain Brauer groups and exploring implications for division algebras.

Contribution

It proves a short exact sequence relating Dec groups and Brauer groups, and characterizes when they are equal for square-free exponents, also providing a counterexample for higher exponents.

Findings

Dec(K/F) equals Br_t(K/F) when t is square-free

Prime exponent division algebras over Henselian valued fields are tensor products of cyclic algebras

Counterexample constructed for higher exponent division algebras

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 that has as a consequence 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.