Rational or irrational, that is the question

There are products between irrational numbers that are rationals and other that are irrationals. What about the specific case of $\frac{\pi}{e}$ ? We need more informations, like the fact that $\pi+e$ and $\frac{\pi}{e}$ do not satisfy any polynomial equation of degree $≤8$ with integer coefficients of average size $10^{9}$ (cf. Numerical Results on the Transcendence of Constants , 1988, by David H. Bailey). But this is still not enough to answer this question.