There are several Galois theories that describe the relations among solutions to functional equations. As relatively easy application of these theories, one can prove that the exponential function e^x is not algebraic, that the Gamma function Gamma(x) does not satisfy any differential equations with polynomial coefficients, and that the incomplete Gamma function gamma(x,t) does not satisfy any algebraic differential equations with respect to d/dt. Although these results are classical, the Galois theoretic point of view gives a more conceptual and formal (i.e., analysis-free) explanation of these facts than many of the earlier proofs, based only on the defining functional equations for these functions: d(e^x)/dx=e^x, Gamma(x+1)=x\Gamma(x), and d^2 gamma/dx^2 (x,t)=(t-1-x)/x d gamma/dx}(x,t). I will give an introduction to these Galois theories and illustrate them by applying them in these concrete examples.