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

Abstract

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