  D. Margalit and S. Schleimer found examples of roots of the Dehn twist about
a nonseparating curve in a closed orientable surface, that is, homeomorphisms
whose nth power is isotopic to the Dehn twist. Our main theorem gives
elementary number-theoretic conditions that describe the values of n for which
an nth root exists, given the genus of the surface. Among its applications, we
show that n must be odd, that the Margalit-Schleimer roots achieve the maximum
value of n among the roots for a given genus, and that for a given odd n, nth
roots exist for all genera greater than (n-2)(n-1)/2. We also describe all nth
roots having n greater than or equal to the genus.
