Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of $p$ belong to the open unit disk in the complex plane (such a polynomial is necessarily irreducible, and is sometimes called a Pisot polynomial).

Is $\mathrm{Gal}(p(X)/\mathbb{Q}) \in \{A_n,S_n\}$ ?

Pisot polynomials are quite common, for instance Perron has shown that if $p(X) = X^n + a_{n-1}X^{n-1} + \dots + a_1X + a_0$ and $|a_{n-1}| > 1 + a_0 + a_1 + \dots + a_{n-2}$ then $p(X)$ is a Pisot polynomial.