  In this paper we will give a global description of the Frobenius for the
division fields of an elliptic curve E which is strictly analogous to the
cyclotomic case. This is then applied to determine the splitting of a prime p
in subfields of such a division field. Such fields include a large class of
non-solvable quintic extensions and our application provides an arithmetic
counterpart to Klein's "solution" of quintic equations using elliptic
functions. A central role is played by the discriminant of the ring of
endomorphisms of the elliptic curve reduced modulo p.
