All dihedral division algebras of degree five are cyclic

Abstract:

In 1982 Rowen and Saltman proved that every division algebra which is split by a dihedral extension of degree $2n$ of the center, $n$ odd, is in fact cyclic. The proof requires roots of unity of order $n$ in the center. We show that for $n=5$, this assumption can be removed. It then follows that ${}_{5\!\!\!\:}\operatorname {Br}(F)$, the $5$-torsion part of the Brauer group, is generated by cyclic algebras, generalizing a result of Merkurjev (1983) on the $2$ and $3$ torsion parts.