Polynomial composites and certain types of fields extensions

In this paper I consider polynomial composites with the coefficients from $K\subset L$. We already know many properties, but we do not know the answer to the question of whether there is a relationship between composites and field extensions. I present the characterization of some known field extensions in terms of polynomial composites. This paper contains the opening problem of characterization of ideals in polynomial composites with respect to various field extensions. I also present the full possible characterization of certain field extensions.