Characterization of field homomorphisms and derivations by functional equations

Summary.

Let K and $\overline K$ be fields containing ${\Bbb Q}$. We characterize pairs of additive functions $f,g: K \to \overline K$ satisfying a functional equation¶¶$g(x^{ln}) = f(x^l)^n \quad \text{respectively} \qquad g(x^{ln}) = Ax^{ln} + x^{ln-l}f(x^l)$,¶where $n \in {\Bbb Z} \setminus \{0,1\}$, $l\in {\Bbb N}$ and $A \in K$.