Basic Proof
To prove the validity of the criterion, suppose Q satisfies the criterion for the prime number p, but that it is nevertheless reducible in ℚ, from which we wish to obtain a contradiction. From Gauss' lemma it follows that Q is reducible in ℤ as well, and in fact can be written as the product Q = GH of two non-constant polynomials G, H (in case Q is not primitive, one applies the lemma to the primitive polynomial Q/c (where the integer c is the content of Q) to obtain a decomposition for it, and multiplies c into one of the factors to obtain a decomposition for Q). Now reduce Q = GH modulo p to obtain a decomposition in (ℤ/pℤ). But by hypothesis this reduction for Q leaves its leading term, of the form axn for a non-zero constant a ∈ ℤ/pℤ, as the only nonzero term. But then necessarily the reductions modulo p of G and H also make all non-leading terms vanish (and cannot make their leading terms vanish), since no other decompositions of axn are possible in (ℤ/pℤ), which is a unique factorization domain. In particular the constant terms of G and H vanish in the reduction, so they are divisible by p, but then the constant term of Q, which is their product, is divisible by p2, contrary to the hypothesis, and one has a contradiction.
Read more about this topic: Eisenstein's Criterion
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