Compositions of polynomials with coefficients in a given field (Q1599109)

The author considers \(F \subset K\) fields of characteristic \(0\) and \(p\), \(q\) polynomials in \(K[x]\) such that \(p\circ q\in F[x]\). He proves that if the leading coefficient and the constant term of \(q\) are both in \(F\), then \(p\) and \(q\) must be in \(F[x]\). The proof is based on a result concerning the \(F\)-deficit \(D_F\) of the composition of the polynomials \(p\) and \(q\), with \(D_F(p) = n-r\), \(n= \deg(p)\) and \(x^r\) the largest power of \(x\) with a coefficient not in \(F\). The paper also contains related results for polynomials with the coefficients in two rings and for polynomials in two variables.