Can the product of only some of the algebraic conjugates be an integer?

Question

Suppose I know that $x_1,\dots, x_n$ are algebraic conjugates and suppose that their product is a rational integer: $$ \prod_{i=1}^{n}x_i\in \mathbb{Z} $$

Is it possible that there exists some other conjugate $x_0$ such that $x_0\neq x_i$ for any $i=1,\dots,n$?

Answer 1

Here is an example.

Let $\zeta$ be a primitive $n$th root of unity. Then $\zeta \cdot \zeta^{-1} = 1$.

Answer 2

$$(\sqrt3+\sqrt2)(\sqrt3-\sqrt2)=1$$