Let $a_0 + a_1 x^1 + a_2 x^2 + \cdots + a_n x^n$ be a polynomial in $\mathbb{Z}[x]$ such that $a_0 \neq 0$ and $|a_i| \le 1$ for $i=0,1,\cdots,n$. Then my conjecture is that this polynomial is separable. Is there any way to prove this, or is this wrong in general? Thanks for your help.

**Edit**
Is there any reason why $g(t) = \sum_{q \text{ prime }, q\le p} t^{q-2}$ for a prime $p\ge 7$ should be separable?

I can prove that any root has absolute value $\le 2$, there exists exactly one negative real root, zero positive real roots and all other complex roots come in conjugate pairs, since the degree is odd.

It seems that those polynomials (except for $p=13$) are also irreducible. I tried to apply Newton polygons to show irreducibility from which separability would follow, but without success.

Any help is highly appreciated.