  Let E/K be an elliptic curve defined over a number field, and let p be a
prime number such that E(K) has full p-torsion. We show that the order of the
p-part of the Shafarevich-Tate group of E/L is unbounded as L varies over
degree p extensions of K. The proof uses O'Neil's period-index obstruction. We
deduce the result from the fact that, under the same hypotheses, there exist
infinitely many elements of the Weil-Chatelet group of E/K of period p and
index p^2.
