Introduction.Let E be an elliptic curve over Q and F the maximal elementary abelian 2-extension of Q, that is, F := Q({ √ m; m ∈ Z}). It is known that the torsion subgroup E(F ) tors of E(F ) is finite (Ribet [8]). More precisely, Laska and Lorenz showed that there exist at most thirty-one possibilities for E(F ) tors (see [3, Theorem] or Theorem 2.1). However, it is not known whether all the groups listed in Theorem 2.1 can happen as E(F ) tors . Now assume that E has non-cyclic torsion over Q; then by Mazur's theorem ([4]), the group E(Q) tors is isomorphic to Z/2Z ⊕ Z/mZ, where m = 2, 4, 6 or 8. Such an elliptic curve has a Weierstrass model E : y 2 = x(x + M )(x + N ), where M and N are non-zero integers with M > N . Further we may assume that the greatest common divisor (M, N ) of M and N is a square-free integer or 1, since for any positive integer d, E is isomorphic over Q to an elliptic curve E d 2 given by y 2 = x(x+d 2 M )(x+d 2 N ) by replacing x with x/d 2 and y with y/d 3 , respectively. Then using the result of Ono ([6, Main Theorem 1], see also Theorem 2.2), Kwon classified the torsion subgroup of E over all quadratic fields ([2, Theorem 1]); Qiu and Zhang classified the torsion subgroup of E for a certain elliptic curve E with E(Q) tors Z/2Z ⊕ Z/2Z over all elementary abelian 2-extensions of Q, i.e., over all number fields of type (2, . . . , 2) ([7, Theorems 3 and 4]); Ohizumi classified the torsion subgroup of E for an elliptic curve E with E(Q) tors Z/2Z ⊕ Z/8Z or Z/2Z ⊕ Z/6Z over all bicyclic biquadratic fields, i.e., over all number fields of type (2, 2) ([5, Main Theorems 4.1 and 4.2]).In this paper, first we completely determine the structure of the torsion subgroup E(F ) tors when E(Q) tors is non-cyclic: