On the second irreducibility theorem of I. Schur

Let \(n\) be a positive integer different from \(8\) and \(n+1 \neq 2^u\) for any integer \(u\geq 2\). Let \(\phi(x)\) belonging to \(Z[x]\) be a monic polynomial which is irreducible modulo all primes less than or equal to \(n+1\). Let \(a_j(x)\) with \(0\leq j\leq n-1\) belonging to \(Z[x]\) be polynomials having degree less than \(\deg\phi(x)\). Assume that the content of \(a_na_0(x)\) is not divisible by any prime less than or equal to \(n+1\). We prove that the polynomial

$$ f(x) = a_n\frac{\phi(x)^n}{(n+1)!}+ \sum _{j=0}^{n-1}a_j(x)\frac{\phi(x)^{j}}{(j+1)!} $$

is irreducible over the field \(Q\) of rational numbers. This generalises a well-known result of Schur which states that the polynomial \( \sum _{j=0}^{n}a_j\frac{x^{j}}{(j+1)!}\) with \(a_j \in Z\) and \(|a_0| = |a_n| = 1\) is irreducible over \(Q\). For proving our results, we use the notion of \(\phi\)-Newton polygons and a few results on primes from number theory. We illustrate our result through examples.