| Title |
Degree of the product of two algebraic numbers one of which is of prime degree / |
| Authors |
Virbalas, Paulius |
| DOI |
10.3390/math11061485 |
| Full Text |
|
| Is Part of |
Mathematics.. Basel : MDPI. 2023, vol. 11, iss. 6, art. no. 1485, p. [1-16].. eISSN 2227-7390 |
| Keywords [eng] |
degree of an algebraic number ; Galois theory ; transitive permutation groups |
| Abstract [eng] |
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. |
| Published |
Basel : MDPI |
| Type |
Journal article |
| Language |
English |
| Publication date |
2023 |
| CC license |
|
