Is every irreducible polynomial reducible after a polynomial substitution? (Q2420380)

In the paper under review, the author investigates the question as the main problem. Let \( K \) be a non-algebraically closed field and \( f\in K[x] \) be an irreducible polynomial of degree \( d\ge 3 \). Does there exist a polynomial \( h\in K[x] \) of degree \( \le d-1 \), such that \( f(h(x)) \) is irreducible in \( K[x] \)? With an illustration with examples, the author is able to resolve the question in the cases \( d=3\) and \( d=4 \). The question is difficult in full generality and the author is unable to obtain the general result. However, during the investigations the author observed some regularities and described them in a short section. Finally, the author states the inverse question of the main problem. Let \(K\) be a non-algebraically closed field and \(h \in K[x]\) of degree \(\ge 2\) be given. Does there exist an irreducible polynomial \(f \in K[x]\) of degree \(>\) deg \(h\), such that the polynomial \(f (h(x))\) is reducible in \(K[x]\)?