The existence of Pisot numbers in any real number field Wikipedia claims that, given a real algebraic number field $K$ of degree $n$, there is an algebraic integer $r \in K$ of degree $n$ such that $r>1$, but every conjugate of $r$ has modulus $<1$ (Such a number is called a Pisot number). I'm trying to prove this fact.
I would be finished if I could show that given $n$ distinct real numbers $a_1,...,a_n$ there is a polynomial $P$ with integer coefficients such that $P(a_1)>1$ and $|P(a_i)|<1$ for $i=2,...,n$. 
Does anyone know whether the above conjecture is valid? Is there another way to prove Wikipedia's proposition? Any suggestion for a possible solution is appreciated.
 A: For $n=1$, this conjecture is trivially true. For $n>1$, you have even more: you can find an $r$ that is a $(n-1)$-hyperbolic unit Pisot number. Let me explain what that all means:
So take $K$ to be a number field of degree $n$ with monomorphisms $\sigma_i:K \rightarrow \mathbb{C}$ for $i \in \{1, \ldots, n\}$. Denote by $\mathcal{O}_K \subset K$ the subring of algebraic integers in $K$ and $U_K$ the group of units in $\mathcal{O}_K$.
An element $\mu \in U_K$ is called $c$-hyperbolic if
$$
\forall k \in \{1, \ldots, c\}, \forall i_1, \ldots, \forall i_k \in \{1, \ldots, n\}:|\sigma_{i_1}(\mu)\cdots\sigma_{i_k}(\mu)|\neq 1.
$$
A proposition from Existence of Anosov diffeomorphisms on
infra-nilmanifolds modeled on free nilpotent Lie
groups:

Let $K$ be a number field of degree $n$ which is not totally imaginary. Then there exists a $c$-hyperbolic $\mu \in U_K$ for all $c \leq n - 1$.

In a remark a bit lower in the same text:

From the proof of Proposition 3.6 it follows that if $K$ is a number field of degree $n$ which is not totally imaginary, then we can always find a $c$-hyperbolic algebraic unit $\mu$ (for any $c < n$)
  such that $|\mu| > 1$, but all other conjugates have absolute value strictly smaller than $1$. So we can assume that $\mu$ is a so called Pisot number.

I know this answer is a bit overkill, but it illustrates that even more is possible in this situation.
