Trace field degrees in the Torelli group

We show that for g\ge 2 , all integers 1\le d\le 3g-3 arise as trace field degrees of pseudo-Anosov mapping classes in the Torelli group of the closed orientable surface of genus g . Our method uses the Thurston–Veech construction of pseudo-Anosov maps, and we provide examples where the stretch factor has algebraic degree any even number between 2 and 6g-6 . This validates a claim by Thurston from the 1980s.