Many people have seen the proof that is irrational (eg e is irrational); fewer have read the proof that is irrational (eg Pi is irrational) but how many have seen the proofs that and are transcendental ie not the solution of any polynomial in integer (or rational) coefficients? It’s a shame as these results are fascinating and are easily accessible to degree level students.
Of course, if you are willing to accept Lindemann’s result that if is algebraic then is transcendental, then the fact that is transcendental follows immediately from the transcendence of (click on read more below if you can’t see why).
But that’s silly as Lindemann’s theorem is hard to prove. However, I have some notes, typed out on green banda sheets. They were a supplement to a course given by Ian Stewart back in the long forgotten past, well, 1970 to be precise. Very good they are too. I have LaTeXed them and you can read the proofs yourself here: The Transcendence of Pi.
Ian Stewart said (and I agree with him) that back in 1970, a student just needed to understand first-year undergraduate analysis to follow the proofs. I wonder if that is still true?
If were algebraic then so would be. It would follow from Lindemann that is transcendental which is silly. So must be transcendental.