Monomial functions, normal polynomials and polynomial equations

In this paper we consider generalized monomial functions f,g:F→C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f, g:{\mathbb {F}}\rightarrow {\mathbb {C}}$$\end{document} (of possibly different degree) that also fulfill f(P(x))=Q(g(x))x∈F,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(P(x))= Q(g(x)) \qquad \left( x\in {\mathbb {F}}\right) , \end{aligned}$$\end{document}where P∈F[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P\in {\mathbb {F}}[x]$$\end{document} and Q∈C[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q\in {\mathbb {C}}[x]$$\end{document} are given (classical) polynomials.