Some functional equations related to number theory

We introduce a new functional equation (E(alpha)), which is originating from the product in the number field . We give an explicit description of the solutions of this equation for and investigate these results to find the solutions of d'Alembert's type and a Van Vleck's functional equations originating from number theory. Our considerations refer to the paper [2] in which L. R. Berrone and L. Dieulefait determine, for a fixed real , the real valued solutions of the equation f(x(1), y(1)) f(x(1), y(1)) = f((x(1)x(2)) + alpha y(1)y(2), x(1)y(2) + x(2)y(1), (x(1), y(1)), (x(2), y(2)) is an element of R-2