Exercise 7.5.2

Proof. Suppose that $f(x,y)$ is irreducible $F[x,y]$. We prove that it is irreducible in $F(y)[x]$, using Gauss’s Lemma:

Theorem A.5.8 : Let $R$be an UFD with field of fractions $K$. Suppose that $f\in R[x]$is non constant and that $f=\mathit{gh}$, where $g,h\in K[x]$. There is a nonzero $\delta \in K$such that $\tilde{g}=\mathit{\delta g}$and $\tilde{h}={\delta }^{-1}h$have coefficients in $R$. Thus $f=\tilde{g}\tilde{h}\in R[x]$.

In the context of the Exercise 7.5.2, take $R=F[y]$, whose field of fractions is $F(y)$.

Suppose that $f=\mathit{gh}$, where $g,h\in F(y)[x]$. By Theorem A.5.8, there exists $\delta \in F(y),\delta \ne 0,$ such that $\tilde{g}=\mathit{\delta g}\in F[y][x]=F[x,y]$ and $\tilde{h}={\delta }^{-1}h\in F[x,y]$. Then $f=\tilde{g}\tilde{h}$, where $f,\tilde{g},\tilde{h}\in F[x,y]$. As $f$ is irreducible in $F[x,y]$, $\tilde{g}\in F^{\text{∗}}$ or $\tilde{f}\in F^{\text{∗}}$. Then $g\in F(y)$ or $h\in F(y)$, which proves the irreducibility of $f$ in $F(y)[x]$.

In particular, $p(x)-\mathit{yq}(x)$, irreducible in $F[x,y]$, is so irreducible in $F(y)[x]$. □