The Existence of the Exponential Function

The purpose of this paperlet is to use some homological algebra in order to prove the existence of a power series $e(x)$ (with coefficients in ${\mathbb {Q} }$) which satisfies the non-linear equation

[Main]

$e(x+y)=e(x)e(y)$

as well as the initial condition

$e(x)=1+x+$(higher order terms).

Alternative proofs of the existence of $e(x)$ are of course available, including the explicit formula $e(x)=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}$. Thus the value of this paperlet is not in the result it proves but rather in the story it tells: that there is a technique to solve functional equations such as [Main] using homology. There are plenty of other examples for the use of that technique, in which the equation replacing [Main] isn't as easy. Thus the exponential function seems to be the easiest illustration of a general principle and as such it is worthy of documenting.