Let K be a field and let E be an elliptic curve defined over K. Let m be a positive integer, prime with char(K) if char(K) = 0; we denote by E[m] the m-torsion subgroup of E and by K m := K(E[m]) the field obtained by adding to K the coordinates of the pointsWe look for small sets of generators for K m inside {x 1 , y 1 , x 2 , y 2 , ζ m } trying to emphasize the role of ζ m (a primitive m-th root of unity). In particular, we prove that K m = K(x 1 , ζ m , y 2 ), for any odd m 5. When m is prime and K is a number field we prove that the generating set {x 1 , ζ m , y 2 } is often minimal. We also describe explicit generators, degree and Galois groups of the extensions K m /K for m = 3 and m = 4, when char(K) = 2, 3. * L.