The Endomorphism Rings of Jacobians of Cyclic Covers of the Projective Line

Suppose K is a eld of characteristic 0, Ka is its algebraic closure, p is an odd prime. Suppose, f(x) 2 K[x] is a polynomial of degree n 5 without multiple roots. Let us consider a curve C : y = f(x) and its jacobian J(C). It is known that the ring End(J(C)) of all Ka-endomorphisms of J(C) contains the ring Z[ p] of integers in the pth cyclotomic eld (generated by obvious automorphisms of C). We prove that End(J(C)) = Z[ p] if the Galois group of f over K is either the symmetric group Sn or the alternating group An.