A Functional Equation Originated from the Product in a Cubic Number Field.

Let K be either R or C and α ∈ R . We determine the solutions f : R 3 → K of the following new parametric functional equation: f (x 1 x 2 + α y 1 z 2 + α y 2 z 1 , x 1 y 2 + x 2 y 1 + α z 1 z 2 , x 1 z 2 + x 2 z 1 + y 1 y 2) = f (x 1 , y 1 , z 1) f (x 2 , y 2 , z 2) , (x 1 , y 1 , z 1) , (x 2 , y 2 , z 2) ∈ R 3 , which results from the product of two numbers in a cubic free field. We equip R 3 with a binary operation to show that the non-zero solutions of this equation are monoid homomorphisms and we investigate our results to introduce and find the solutions of d'Alembert's functional equations with endomorphisms. [ABSTRACT FROM AUTHOR]