It is well known to all of us that the rational powers of $e$ are irrational numbers. Many of the proofs proving this use a similar approach as proving $e$ irrational using Niven's Polynomials. Is it true that rational powers of $e$ are also transcendental numbers using proofs similar(as proved by Hermite) for proving $e$ transcendental?
How to measure the irrationality measure of those rational powers of $e$?
In field theory language: We have $\mathbb{Q}(e^{m/n})\subseteq\mathbb{Q}(e^{1/n})$ so WLOG we may consider $e^{1/n}$. Observe $\mathbb{Q}(e^{1/n})/\mathbb{Q}(e)$ is finite and $\mathbb{Q}(e)/\mathbb{Q}$ is infinite, so $\mathbb{Q}(e^{1/n})/\mathbb{Q}$ is infinite: $e^{1/n}$ is transcendental.