# Why can't $\alpha - 1$ be a unit here?

I've been reading a the book "Distribution Modulo one and Diophantine Approximation" by Yann Bugeaud. Bugeaud is proving a statement due to Toufic Zaïmi about Salem numbers $$\alpha$$. A Salem number is an algebraic integer such that one of its Galois conjugates is $$\frac{1}{\alpha}$$ and all of the other Galois conjugates lie on the unit circle.

Bugeaud shows that for Salem numbers $$\mathbb{Q}$$ that satisfy a certain additional property, we have that $$|P(1)| \neq 1$$, where $$P$$ is the minimal polynomial of $$\alpha$$ over $$\mathbb{Q}$$. Bugeaud concludes that $$\alpha - 1$$ cannot be a unit. I don't understand where this conclusion comes from. Is this a general fact about algebraic integers?

More specifically, if $$\alpha$$ is an algebraic integer with minimal polynomial $$P$$, and $$\alpha - 1$$ is a unit, is it necessarily true that $$|P(1)| = 1$$? If so, why?

Note Let $$P(X)$$ be the minimal Polynomial of $$\alpha$$. Then $$Q(X)=P(X+1)$$ is irreducible and satisfies $$Q(\alpha-1)=0$$. Therefore, $$Q$$ is teh minimal polynomial of $$\alpha-1$$.
Now, $$\alpha-1$$ is an unit if and only if $$|Q(0)|=|P(1)|=1$$.
The norm of $$\alpha-c$$, where $$c\in\Bbb Z$$ is $$(-1)^dP(c)$$ where $$d=\deg P$$. If $$\alpha-1$$ is a unit, then its norm is $$\pm1$$ so $$P(1)=\pm1$$.