To which statement is Artin referring in this proof about factoring integer polynomials?

Question

The following is an excerpt in Artin's Algebra (2nd ed., pg. 371)

We pose the problem of factoring an integer polynomial $$ f(x)=a_nx^n+\cdots+a_1x+a_0, $$ with $a_n\neq 0$. Linear factors can be found fairly easily.

Lemma 12.4.2b

A primitive polynomial $b_1x+b_0$ divides $f$ in $\mathbb{Z}[x]$ if and only if the rational number $-b_0/b_1$ is a root of $f$.

Proof.

(b) According to Theorem $\color{red}{12.3.10(c)}$, $b_1x+b_0$ divides $f$ in $\mathbb{Z}[x]$ if and only if it divides $f$ in $\mathbb{Q}[x]$, and this is true if and only if $x+b_0/b_1$ divides $f$, i.e., $-b_0/b_1$ is a root.

On the same page, Theorem 12.3.10 does not have a part (c).

Theorem 12.3.10

If $R$ is a unique factorization domain, the polynomial ring $R[x_1,\ldots,x_n]$ in any number of variables is a unique factorization domain.

Does anyone know about this typo? If so, to which statement is Artin referring?

Answer 1

Let's take a look at the following piece of the proof of Lemma 12.4.2b.

$b_1x+b_0$ divides $f$ in $\mathbb{Z}[x]$ if and only if it divides $f$ in $\mathbb{Q}[x]\quad(1)$

Theorem 12.3.6(a) on pg. 369 says

Let $f_0$ be a primitive polynomial, and let $g$ be an integer polynomial. If $f_0$ divides $g$ in $\mathbb{Q}[x]$, then $f_0$ divides $g$ in $\mathbb{Z}[x]$.

$(\Leftarrow)$ holds for $(1)$ by the above theorem since $b_1x+b_0$ is primitive by the statement of the lemma; however, $(\Rightarrow)$ does not follow from that theorem. To get $(\Rightarrow)$, we know that if $b_1x+b_0$ divides $f$ in $\mathbb{Z}[x]$, it also divides $f$ in $\mathbb{Q}[x]$.