Are \(p\)-algebras having cyclic quadratic extensions necessarily cyclic? (Q1291116)

In his efforts to find a non-cyclic \(p\)-algebra of prime degree \(p\), \textit{A. A. Albert} [in J. Algebra 5, 110-132 (1967; Zbl 0144.02503)] considered a class of \(p\)-algebras which become cyclic under quadratic extension of the ground field. The author places Albert's examples in a somewhat more general setting and proves that any \(p\)-algebra of degree \(p\) and satisfying a certain condition (which includes Albert's examples), which becomes cyclic after a quadratic extension of the centre must already have been cyclic. He goes on to describe a generic presentation of \(p\)-algebras which become cyclic after a quadratic ground field extension and raises the question whether non-cyclic \(p\)-algebras can be obtained this way. Specifically, he constructs an algebra which is the most general division \(p\)-algebra of degree \(p\) which becomes a symbol after quadratic extension. He observes that for \(p>3\), either this is a non-cyclic division \(p\)-algebra of degree \(p\), or every division \(p\)-algebra of degree \(p\) which becomes cyclic after a quadratic extension must be cyclic.