It is not difficult to recognize Salem numbers of degree 6. Since is reciprocal, we can writewhere The zeros of U are the numbers and hence U must have two roots in the open interval -2 < x < 2 and one root with x > 2. This is equivalent to the following requirements: (i) , (ii) has real roots, the smaller of which, satisfies , and . We used this criterion in compiling our list of Salem numbers.
We also require U to be irreducible. Since U is cubic, it suffices that for any integer n. Only a finite set of n need be checked since, by a well-known estimate of Cauchy, for . It is easy to see that P can only factor into factors of even degree since the roots and must belong to the same factor, or else there would be a factor the product of whose roots is in absolute value less than 1, which is clearly impossible. Thus, P is irreducible if and only if U is irreducible. The following useful result is an elementary deduction from the above discussion.
Let P be as in (2.1) and U as in (2.3). A necessary condition for P to be the minimal polynomial of a Salem number is that for . A sufficient condition for P to be the minimal polynomial of a Salem number is, in addition to (2.4), any one of , or .
In general, if P is the minimal polynomial of a Salem number of degree d = 2s, then there is a monic polynomial U with integer coefficients for which . The numbers are the zeros of U repeated twice each. The following computation gives a useful expression for :