Representing classes in the Brauer group of quadratic number rings as smash products (Q1079637)

\textit{J. Gamst} and \textit{K. Hoechsmann} [C. R. Acad. Sci., Paris, Sér. A 269, 560-562 (1969; Zbl 0197.032)] showed that the smash product of Galois objects with respect to a dual pair of finite Hopf R-algebras is an Azumaya R-algebra. This paper describes examples of this construction for rank 4 Azumaya algebras over rings of integers of number fields. In case the Hopf algebras are free of rank 2 and the Galois objects have normal bases, the resulting smash products are generated as R-algebras by z and w with \(z^ 2-az-e=0\), \(w^ 2-bw-d=0\), \(zw+wz=bz+aw-1\), where a,b,d,e are in R with \(ab=2\). The two non-isomorphic, Brauer equivalent rank 4 Azumaya algebras over the ring of integers of a real quartic field, found by \textit{R. G. Swan} [Ann. Math., II. Ser. 76, 55-61 (1962; Zbl 0112.027)] are smash products of this kind. All such smash products representing the non-trivial class in the Brauer group of R, R the ring of integers of \({\mathbb{Q}}(\sqrt{m})\), m squarefree, positive, and \(<100\), are found. The non-trivial class of the Brauer group of the ring of integers of \({\mathbb{Q}}(\sqrt{5})\) is shown to be not representable as a smash product of rank 2 Galois objects.