A Salem number with degree 34 and trace −3

In this paper, we give a new Salem number with degree 34 and trace −3, whereas the Salem number of lower degree known up to now, with trace −3, was of degree 54. The method used to find this Salem number also provides a list of 41 monic integer polynomials of smallest possible trace, irreducible, with degree 17, and having all roots real and positive. Moreover, we think that this list may be complete.