TITLE: Salem Numbers, roots on the unit circle
QUESTION [8 upvotes]: There are algebraic integers which are not roots of unity , for example consider the irreducible polynomial $ P(x)= x^4-2x^3-2x+1 $.
A computer software can show that this polynomial has two real roots outside the unit circle(one greater than one and the other less than one) and two roots on the unit circle. However I don't know how to prove this rigorously that there are two roots on the unit circle. 
Usually when I wanted to prove some root of a polynomial is on the unit circle , I'd multiply that by some other polynomial to get something of the form $ x^n - 1 $ and it's obvious that every root of such expression has norm one, however , in this case this is not possible since none of roots of $ P(x) =0 $ are roots of unity. 
Actually there is a whole lot of examples, called Salem numbers. It's an algebraic integer $\lambda > 1$ such that all of its Galois conjugates are on the unit circle except $\frac{1}{\lambda}$. The polynomial given above was an example of a minimal polynomial of a Salem number.   
Does anyone have any idea how can I prove this, i.e. roots are on the unit circle except two of them?(I'm looking for a method that can be applied to more than just one example, hopefully lots of Salem numbers)

REPLY [2 votes]: Here is the best way to count roots on the unit circle. 
Your quartic polynomial has even degree and symmetric coefficients: $c_k = c_{4-k}$ for $0 \leq k \leq 4$.  From the symmetry in the coefficients, your polynomial can be written in the form $x^2g(x + 1/x)$ for a polynomial $g(x)$, where in fact $g(x) = x^2 - 2x - 2$.  
The mapping $z \mapsto z + 1/z$ sends the unit circle in a 2-to-1 way onto the interval $[-2,2]$ (except for being 1-to-1 at the endpoints). The roots of your polynomial on the unit circle (which don't include $\pm 1$) are in a 2-to-1 correspondence with roots of $g(x)$ in the interval $[-2,2]$. There is one such root for $g(x)$, so your polynomial has two roots on the unit circle.
Further examples are discussed at https://kconrad.math.uconn.edu/blurbs/galoistheory/numbersoncircle.pdf, including polynomials that don't have symmetric coefficients. Your example is treated at the very start by a more elementary method that does not easily generalize to higher degree and also in later examples by methods that do generalize to higher degree.