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).



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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.



Does the Galois group of a Pisot polynomial contain the alternating group?
David Speyer has beaten me to it, but for what it's worth, a simple explicit example is
$$p(X)=X^4-2X^3-5X^2-4X-1$$
which is a Pisot polynomial, and has Galois group the dihedral group of order 8.


How did I find it? I started from observing that if $\vartheta$ is the positive root of $X^2-X-1$, then $\alpha=1+\sqrt{\vartheta}$ is an algebraic unit with only one conjugate inside the closed unit disk, and thus $-1/\alpha$ is a Pisot-Vijayaraghavan number, with this minimal polynomial.


**Addendum:** The Pisot polynomial


$$p(X)=X^4-2X^3+X-1$$


again has dihedral Galois group of order 8, corresponding to another quadratic extension of $\mathbb{Q}(\vartheta)$, and produces a root less than $2$ (which also happens to be the fourth smallest limit point of the set of PV numbers). This number is mentioned on page 1251 of David W. Boyd, [*Pisot and Salem numbers in intervals of the real line*](http://www.ams.org/journals/mcom/1978-32-144/S0025-5718-1978-0491587-8/S0025-5718-1978-0491587-8.pdf), Math. Comp. **32** (1978), 1244-1260.


