Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree

Let α and β be two algebraic numbers such that deg(α)=m deg(β)=p, where p is a prime number not dividing m. This research focused on the following objectives: to discover new conditions under which deg(αβ)=mp; determine complete list of values deg(αβ) can take. With respect first question, we find if minimal polynomial over Q neither xp+c nor x2+cx+c2, then necessarily deg(αβ)=mp αβ primitive element Q(α,β). supplements some earlier results by Weintraub. second p>2 p−1 divides m, for every divisor k p−1, there exist deg(αβ)=mp/k.