Growth of Selmer groups in non-cyclic Galois extensions

Let E/Q be an elliptic curve, and let p be a prime number. Then it is known in many cases, and expected
to be true in all cases, that there exists a cyclic degree p extension K/Q such that the p-Selmer group of E/K
has the same cardinality as that of E/Q. In this talk I will explain that as soon as one replaces "cyclic degree
p" by even the smallest non-cyclic Galois extensions, this is far from true. Concretely, I will sketch a proof
of the following: there exist elliptic curves E over Q such that for _all_ prime numbers p, and for all extensions
K that are either bi-cyclic of degree p^{^2} or dihedral of degree 2p, the p-Selmer group of E/K is bigger than
that of E/Q.