  Let E be an elliptic curve over Q, and let n=>1. The central object of study
of this article is the division field Q(E[n]) that results by adjoining to Q
the coordinates of all n-torsion points on E(Q). In particular, we classify all
curves E/Q such that Q(E[n]) is as small as possible, that is, when
Q(E[n])=Q(zeta_n), and we prove that this is only possible for n=2,3,4, or 5.
More generally, we classify all curves such that Q(E[n]) is contained in a
cyclotomic extension of Q or, equivalently (by the Kronecker-Weber theorem),
when Q(E[n])/Q is an abelian extension. In particular, we prove that this only
happens for n=2,3,4,5,6, or 8, and we classify the possible Galois groups that
occur for each value of n.
