Perron's criterion 不是“整数多项式” 的反例?

hbghlyj

整数多项式讲的Perron's criterion, 能否举出一个$f(x)\in\Bbb Q[x]\setminus\Bbb Z[x]$定理不成立的例子?
Perron's criterion: Suppose $f(x)=x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}\in\Bbb Z[x]$, where $a_{0}\neq 0$. If either of the following two conditions applies:

• $|a_{n-1}|>1+|a_{n-2}|+\cdots +|a_{0}|$
• $|a_{n-1}|=1+|a_{n-2}|+\cdots +|a_{0}|,\quad f(\pm 1)\neq 0$

then $f$ is irreducible over $\Bbb Z$ (and by Gauss's lemma also over $\Bbb Q$).

hbghlyj

$f(x)=x^\color{#f00}2+\frac 52x+1=(x+\frac 12)(x+2)$