  The $(u,v)$-Pad\'e approximation to a function $f$ is the (unique, up to
scaling) rational approximation $f(x) = P(x)/Q(x) + O(x^{u+v+1})$, where $P$
has degree $u$ and $Q$ has degree $v$. Motivated by recent work of Molin,
Pazuki, and Rabarison, we study the arithmetic of the Pad\'e approximants of
the exponential polynomials. By viewing the approximants as certain Generalized
Laguerre Polynomials, we determine the Galois groups of the diagonal
approximants and prove some special cases of irreducibility.
