×

Reducibility of polynomials after a polynomial substitution. (English) Zbl 1449.12001

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 \( g \in \mathbb{Q}[x]\).

MSC:

12D05 Polynomials in real and complex fields: factorization
11D25 Cubic and quartic Diophantine equations
11D61 Exponential Diophantine equations

Citations:

Zbl 1435.11068
PDFBibTeX XMLCite
Full Text: DOI