  We show that there are Salem numbers of every trace. The nontrivial part of
this result is for
  Salem numbers of negative trace. The proof has two main ingredients. The
first is a novel construction, using pairs of polynomials whose zeros interlace
on the unit circle, of polynomials of specified negative trace having one
factor a Salem polynomial, with any other factors being cyclotomic.
  The second is an upper bound for the exponent of a maximal torsion coset of
an algebraic torus in a variety defined over the rationals.
  This second result, which may be of independent interest, enables us to
refine our construction to avoid getting cyclotomic factors, giving a Salem
polynomial of any specified trace, with a trace-dependent bound for its degree.
We show also how our interlacing construction can be easily adapted to produce
Pisot polynomials, giving a simpler, and more explicit, construction for Pisot
numbers of arbitrary trace than previously known.

Number Theory