A note on the \(p\)-divisibility of resultants (Q2785639)

Let \(p\) be a prime and let \(f(x)\) be a monic polynomial with coefficients in \(\mathbb{Z}\). Under a well-known condition there is a unique sequence \((a_k)\) in a fixed residue class mod \(p\) such that \(f(a_k) \equiv 0 \pmod {p^k}\) and \(0\leq a_k < p^k\) (\(k=1,2,\dots\)). Let \(g(x) \in \mathbb{Z}[x]\) be another monic polynomial with a corresponding sequence \((b_k)\). Assume that \(a_h=b_h\) for some \(h\geq 1\). The authors prove that the resultant \(R(f,g)\) is divisible by \(p^h\). More generally, if there are several such pairs of sequences (each pair belonging to a different residue class mod \(p\)), then \(R(f,g)\) is divisible by the product of the corresponding \(p^h\). As an application the authors take \(f(x) = x^{p-1}-1\) and \(g(x) = (x+1)^{p-1}-1\) and obtain a criterion, in terms of Wendt's determinant, about the nonsolvability of Fermat's equation (*) \(x^p+y^p=z^p\) in the first case. This result, which improves upon a previous result, is based on a known relationship between the solvability of (*) and the zeros mod \(p^2\) of \(f(x)\).