Drungilas, Paulius; Dubickas, Arturas Reducibility of polynomials after a polynomial substitution. (English) Zbl 1449.12001 Publ. Math. Debr. 96, No. 1-2, 185-194 (2020). Let \(K\) be a field, and let \(f \in K[x]\) be a polynomial of degree \(d \geq 3\) which is irreducible over \(K\). M. Ulas [J. Number Theory 202, 37–59 (2019; Zbl 1435.11068)] raised the problem of the existence of a polynomial of degree \(\leq d-1\) such that the composition polynomial \(f(g(x))\) is reducible in \(K\). He proved the existence of such polynomial in case of \(d\leq 4\). Here the authors solve the above problem.There is an integer \(\ell\) in the range \(2 \leq \ell\leq d-1\) and a polynomial \(h\in K[x]\) of degree \(d\ell\) such that \(f(h(x))\) of degree \(d\ell\) is reducible in \( K[x].\) In particular, for any \(K\) and \(f\) as above, there is \(h\in K[x]\) of degree \(\ell= d-1\) such that \(f(h(x))\) of degree \(d(d-1)\) has an irreducible factor \(f^{*}(x) := x^df(x^{-1}) \in K[x]\) of degree \(d.\)It is also shown that for any non-constant polynomial \(g \in K[x],\) the polynomial \(f(g(x))\) is irreducible over \( K\) if and only if for some root \(\alpha\) of \(f\) the polynomial \(g(x)-\alpha\) is irreducible over \(K(\alpha)\).The authors also characterize all quartic polynomials \(f \in K[x]\), where \(K\) is a field of characteristic zero, for which \(f(g(x))\) remains irreducible over \(K\) under any quadratic substitution \(g \in K[x]\). This characterization is given in terms of \(K\)-rational points on an elliptic curve of genus 1.As a corollary, they prove that the polynomial \(g(x)^4 + 1\) is irreducible over \(\mathbb{Q}\) for any quartic polynomial \( g \in \mathbb{Q}[x]\). Reviewer: Piroska Lakatos (Debrecen) Cited in 2 Documents MSC: 12D05 Polynomials in real and complex fields: factorization 11D25 Cubic and quartic Diophantine equations 11D61 Exponential Diophantine equations Keywords:irreducible polynomial; composition of polynomials; field; exponential Diophantine equations; quartic polynomial; elliptic curve Citations:Zbl 1435.11068 PDFBibTeX XMLCite \textit{P. Drungilas} and \textit{A. Dubickas}, Publ. Math. Debr. 96, No. 1--2, 185--194 (2020; Zbl 1449.12001) Full Text: DOI