On Diamond Products Ensuring Irreducibility of the Associated Composed Product

Let f and g be two irreducible polynomials of coprime degrees m and n whose zeroes lie in a set G⊆F¯q. Let ⋄ be a diamond product on G. We define the weaker cancelation property of ⋄ and show that it is sufficient to conclude that the composed product of f and g derived from ⋄ is an irreducible polynomial of degree mn. We also prove that a wide class of diamond products on finite fields satisfy the weaker cancelation property. These results extend the corresponding results of Brawley and Carlitz (1987), and Munemasa and Nakamura (2016).