When can the primitive element be written as a sum of two algebraic elements adjoined to the field of the rational numbers
Given a field F and elements α and β not in F, then F(α, β) is the smallest field containing α,β, and F. A simple extension is a field extension which is generated by the adjunction of a single element. The Primitive Element Theorem says that if F is a field of characteristic 0, and α and β are algebraic over F, then there is an element γ in F(α ,β ) such that F(α ; β ) = F(γ). When can we say that γ=α+β? We will introduce some situations where γ=α+β is true and some when this is not true, where F is the field of rational numbers Q.^
Moussa, Mohamad Medhat, "When can the primitive element be written as a sum of two algebraic elements adjoined to the field of the rational numbers" (2015). ETD Collection for University of Texas, El Paso. AAI1600335.