Exercise 10, § 3

Let K be a field and P a polynomial in the ring $K[X, Y]$ in two indeterminates. Suppose that there exists a polynomial $H \in K[X]$ and two polynomials Q, R $\in K[X, Y]$ such that $H(X)P(X, Y) = Q(X, Y)R(X, Y)$ and that the coefficients of $Q(X, Y)$, qua polynomial in Y, are polynomials in $K[X]$ without non-constant common factor. Show that all the coefficients of $R(X, Y)$, qua polynomial in Y, are divisible by $H(X)$. (Consider H, P, Q, R as polynomials in X over the field $K(Y)$ and decompose them into factors of the first degree in $\Omega[X]$ where $\Omega$ is an algebraic closure of $K(Y)$; observe that if $H(X)$ and $Q(X, Y)$ had a factor of the first degree in common, this factor would divide all the coefficients of $Q(X, Y)$, qua polynomial in Y, and use the fact (IV, p. 12) that if $K'$ is an extension of K, the GCD in $K'[X]$ of two polynomials of $K[X]$ belongs to $K[X]$.) \*