this is really an overkill proof: suppose then but, by Eisenstein's criterion, is irreducible over and thus contradiction!
Haha, that just gave me an idea. You know in math we try to find more and more elementary proofs. Well, how about we do the opposite for a change. Prove that is irrational using the most overblown results from math that you can think of, or prove irrationality using a very long and confusing proof.