Linear independence of the numbers $\{1=?iso-8859-1?Q?=2Ce=2Ce^2=2Ce^3\}$_=96_math.stackexchange.com?=

Linear independence of the numbers $\{1,e,e^2,e^3\}$ – math.stackexchange.com

Does someone know a proof that $\{1,e,e^2,e^3\}$ is linearly independent
over $\mathbb{Q}$? The proof should not use that $e$ is transcendental.
$e:$ Euler's number $\{1,e,e^2\}$ is linearly ...