The Brumer-Stark conjecture in some families of extensions of specified degree

As a starting point, an important link is established between Brumer's conjecture and the Brumer-Stark conjecture which allows one to translate recent progress on the former into new results on the latter. For example, if $K/F$ is an abelian extension of relative degree $2p$, $p$ an odd prime, we prove the $l$-part of the Brumer-Stark conjecture for all odd primes $l\ne p$ with $F$ belonging to a wide class of base fields. In the same setting, we study the $2$-part and $p$-part of Brumer-Stark with no special restriction on $F$ and are left with only two well-defined specific classes of extensions that elude proof. Extensive computations were carried out within these two classes and a complete numerical proof of the Brumer-Stark conjecture was obtained in all cases.