ABSTRACT
Let F be a field and let D be an F-central division algebra. A well known theorem of Wedderburn (1921) says that if an irreducible polynomial f(x) ∈ F[x] has a root θ in D then f(x) decomposes into a product of linear factors in D[x]. More precisely there are elements θ 1 = θ, θ 2,…,θk in D, such that each θi is conjugate to θ and f(x) = (x − θk )(x − θ k−1) … (x − θ 2)(x − θ 1). Note: The elements θi need not commute! In the same paper Wedderburn proved that if the dimension of D over F is nine, then one can do better. In that case if θ is in D − F then f(x) (which will be of degree three) decomposes over D in a very special way: there is an element y in D such that y 3 ∈ F x and f(x) = (x − y −2 θy 2)(x − y −1 θy)(x − θ). Wedderburn used this fact to prove that every division algebra of dimension nine is cyclic, that is has a maximal subfield (necessarily of degree three over F) that is a cyclic extension of F.
