Galois theory of Salem polynomials

Let f(x) 2 Z[x] be a monic irreducible reciprocal polynomial of degree 2d with roots r1, 1=r1, r2, 1=r2, . . . , rd, 1=rd. The corresponding trace polynomial g(x) of degree d is the polynomial whose roots are r1 +1=r1, . . . , rd +1=rd. If the Galois groups of f and g are Gf and Gg respectively, then Gg = Gf=N , where N is isomorphic to a subgroup of Cd 2 . In a naive sense, the generic case is Gf = Cd 2 oSd, with N = Cd 2 and Gg = Sd. When f(x) has extra structure this may be re ected in the Galois group, and it is not always true even that Gf = N oGg. For example, for cyclotomic polynomials f(x) = n(x) it is known that Gf = N o Gg 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) 2 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 Gf = N oGg, and moreover that N = Cd 2 or Cd 1 2 , with the latter only possible if d is odd.