Algebraic independence of some values of the exponential function

We prove general results concerning the algebraic independence of three values of the exponential function. For $\beta$ algebraic and of degree 7 and $\alpha$ algebraic and $\neq0,\,1$ there exist among the numbers $\alpha^\beta,\dots,\alpha^{\beta^6}$ three which are algebraically independent. The proof employs a method due to A. O. Gel'fond and N. I. Fel'dman.