Galois theory of Salem polynomials

Abstract

Let f (x) is an element of Z[x] be a monic irreducible reciprocal polynomial of degree 2d with roots r(1), 1/r(1), r(2), 1/r(2), ..., r(d), 1/r(d). The corresponding trace polynomial g(x) of degree d is the polynomial whose roots are r(1) + 1/r(1), ..., r(d) + 1/r(d). If the Galois groups of f and g are G(f) and G(g) respectively, then Gg congruent to G(f)/N, where N is isomorphic to a subgroup of C-2(d). In a naive sense, the generic case is G(f) congruent to C-2(d) x S-d, with N congruent to C-2(d) and G(g) congruent to S-d. When f (x) has extra structure this may be reflected in the Galois group, and it is not always true even that G(f) congruent to N x G(g). For example, for cyclotomic polynomials f (x) = Phi(n)(x) it is known that G(f) congruent to N x G(g) if and only if n is divisible either by 4 or by some prime congruent to 3 modulo 4. In this paper we deal with irreducible reciprocal monic polynomials f (x) is an element of Z[x] that are 'close' to being cyclotomic, in that there is one pair of real positive reciprocal roots and all other roots lie on the unit circle. With the further restriction that f (x) has degree at least 4, this means that f (x) is the minimal polynomial of a Salem number. We show that in this case one always has G(f) congruent to N x G(g), and moreover that N congruent to C-2(d) or C-2(d-1), with the latter only possible if d is odd.