Kummer Elements and the mod-3 Invariant of Albert Algebras

Abstract

Let k be a field with characteristic not 2 and 3. Assume that k contains the cube roots of unity. Let J be a Tits'-first-construction Albert division algebra over k. In this paper we relate Kummer elements in J with the mod-3 invariant g 3 J . We prove that if x ∈ J is a Kummer element with x 3 = λ, then J J D λ for some D, a degree-3 central division algebra over k. We show that if J 1 = J A µ and J 2 = J B ν are Tits' first-construction Albert division algebras withAcademic Press