On the Galois Group of the Generalized Fibonacci Polynomial

be the generalized Fibonacci polynomial. There are a few papers in the literature in which the distribution of the roots of fn is considered. For example, from [5] and [6] we know that there exists a unique positive root θ of fn, which is larger than 1, and all the other roots of fn have modulus less than 1. Moreover, if n is odd, then fn has only one real root, while if n is even, fn has exactly two real roots (the root θ > 1 and one other root in the interval (−1, 0)). For an analysis of the real roots of f ′ n and f ′′ n see [3]. Since fn has a unique positive root θ which is larger than 1 and all the other roots are in the open unit disk, it follows that θ is a Pisot number and fn is a Pisot polynomial. In particular, fn is irreducible. This observation is due to Boyd (see [9]). Thus, we may denote the roots of fn by θ (i) for i = 1, 2, . . . , n, with the usual convention that θ = θ. We also let n = r1 + 2r2, where r1 is the number of real roots of fn and 2r2 is the number of complex non-real roots of fn. From the above comments we know that r2 = ⌊