I don't think anyone mentioned Eisenstein's classic proof. [This presentation][1] of it is pretty good. I find it clear and attractive, especially because it sort of avoids Gauss' Lemma (which is a clever gadget but somehow off-putting). 


  [1]: https://web.archive.org/web/20130628175852/http://math.nmsu.edu:80/~history/eisenstein.pdf