## Proof in Hardy's book

Hey guys,

I've been reading a proof of the transcendence of pi in G.H Hardy's book "An introduction to the theory of numbers" pp 173-176 (see here, and there are lemmas for which some elements of the proof are depenedent on, on pp 170-172), and I'm quite confused about a point in the proof. He ends with "for sufficiently large p". I think this p is distinctly different from the one he defines on page 174. Can anyone give an explanation as to why he has stated "for all sufficiently large p", and exactly what this "new" p is?

Another question on the same proof, I'm also quite confused by his statement on the bottom of page 174; "Hence any such function is a symmetric integral polynomial in $d\omega_1, d\omega_2, \ldots, d\omega_m$"; why is it an integer?

Thank you all so much -- I'm finding Hardy to be quite mystical in the way he writes (not always a good thing! )

Regards,
HTale.