Exercise 13.4.6

Proof.

(a)

Assume that $f\in \mathbb{Z}[x_1,...,x_n,y]$ is irreducible and nonconstant.

Suppose that $f=\mathit{gh}$, where $g,h$ are in $\mathbb{Q}[x_1,\dots ,x_n,y]$. There are positive integers $r,s$ such that $\mathit{rg},\mathit{sh}\in \mathbb{Z}[x_1,\dots ,x_n,y]$ (it is not needed to take $r,s$ as small as possible). Then $\mathit{rsf}=(\mathit{rg})(\mathit{sh})=g_1{h}_1$, where $g_1=\mathit{rg},h_1=\mathit{sh}$ are in $\mathbb{Z}[x_1,\dots ,x_n,y]$.

Write $\mathit{rs}=p_1^{a_1}\cdots p_r^{a_r}$ the decomposition of $\mathit{rs}$ in prime integers. Note that every prime in $\mathbb{Z}$ is irreducible in $\mathbb{Z}[x_1,\dots ,x_n]$, so that

$$\mathit{rsf}=p_1^{a_1}\cdots p_r^{a_r}f$$

is a decomposition in irreducible factors in $\mathbb{Z}[x_1,\dots ,x_n]$. The only units of this UFD are $\pm 1$, so that the unicity of the decomposition in irreducible factors in $\mathbb{Z}[x_1,\dots ,x_n]$ shows that the decompositions of $g_1=\mathit{rg},h_1=\mathit{sh}$ are in the form

$$g_1=\pm p_1^{b_1}\cdots p_r^{b_r}f,h_1=\pm p_1^{c_1}\cdots p_r^{c_r},$$

or

$$g_1=\pm p_1^{b_1}\cdots p_r^{b_r},h_1=\pm p_1^{c_1}\cdots p_r^{c_r}f.$$

Therefore $h_1$ or $g_1$ is in $\mathbb{Z}$, thus $g$ or $h$ is in $\mathbb{Q}$, so is a unit in $\mathbb{Q}[x_1,\dots ,x_n,y]$. This proves that $f$ is irreducible in $\mathbb{Q}[x_1,\dots ,x_n,y]$.

(b)

Since $\mathbb{Z}[x_1,...,x_n,y]$ is UFD, $s_u(y)$ can be uniquely factorized $s_u(y)=h_1\cdots h_n$, where $h_1,...,h_n\in \mathbb{Z}[x_1,...,x_n,y]$ and are irreducible. By the part (a), $h_1,...,h_n$ are irreducible when regarded as elements of $\mathbb{Q}[x_1,...,x_n,y]$. Hence for any irreducible factor $h\in \mathbb{Q}[x_1,...,x_n,y]$ of $s_u(y)$, there exists $i$ such that $h\mid h_i$. It follows that $h$ is associate to $h_i$, $h=\lambda h_i,\lambda \in \mathbb{Q}$, is the product of $h_i$ by a suitable constant from $\mathbb{Q}$, where $h_i$ is a factor of $s_u(y)$ irreducible in $\mathbb{Z}[x_1,...,x_n,y]$.