Solving the Pell equation via R\'edei rational functions

In this paper, we define a new product over ${\mathbb{R}}^{\mathrm{\infty }}$, which allows us to obtain a group isomorphic to ${\mathbb{R}}^{\ast }$ with the usual product. This operation unexpectedly offers an interpretation of the R'edei rational functions, making more clear some of their properties, and leads to another product, which generates a group structure over the Pell hyperbola. Finally, we join together these results, in order to evaluate solutions of Pell equation in an original way.