Eisenstein's criterion of irreducibility simple proof

Question

I'd like to find a proof (as simple as possible) of Eisenstein's criterion of irreducibility. I found here the proof in the picture below. This is rather elegant, yet I do not understand why $g,h$ must have integer coefficients.

Edit:

I found this paper, stating

If a polynomial with integer coefficients is reducible over Q, then it is reducible over Z.

A proof of this is not given though

Answer 1

This has to do with Gauss's Lemma for polynomials. It states that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers.

So in the given proof, they assume that it is reducible over integers to achieve a contradiction in order to prove that it is irreducible over the integers, and hence also over the rationals.