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 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 ( α β ) = m p ; to determine the complete list of values deg ( α β ) can take. With respect to the first question, we find that if the minimal polynomial of β over Q is neither x p + c nor x 2 + c x + c 2 , then necessarily deg ( α β ) = m p and α β 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 ( α β ) = m p / k .