Fields generated by torsion points of elliptic curves

Let K   be a field of characteristic char(K)≠2,3char(K)≠2,3 and let EE be an elliptic curve defined over K. Let m   be a positive integer, prime with char(K)char(K) if char(K)≠0char(K)≠0; we denote by E[m]E[m] the m  -torsion subgroup of EE and by Km:=K(E[m])Km:=K(E[m]) the field obtained by adding to K   the coordinates of the points of E[m]E[m]. Let Pi:=(xi,yi)Pi:=(xi,yi) (i=1,2i=1,2) be a ZZ-basis for E[m]E[m]; then Km=K(x1,y1,x2,y2)Km=K(x1,y1,x2,y2). We look for small sets of generators for KmKm inside {x1,y1,x2,y2,ζm}{x1,y1,x2,y2,ζm} trying to emphasize the role of ζmζm (a primitive m  -th root of unity). In particular, we prove that Km=K(x1,ζm,y2)Km=K(x1,ζm,y2), for any odd m⩾5m⩾5. When m=pm=p is prime and K   is a number field we prove that the generating set {x1,ζp,y2}{x1,ζp,y2} is often minimal, while when the classical Galois representation Gal(Kp/K)→GL2(Z/pZ)Gal(Kp/K)→GL2(Z/pZ) is not surjective we are sometimes able to further reduce the set of generators. We also describe explicit generators, degree and Galois groups of the extensions Km/KKm/K for m=3m=3 and m=4m=4.