  We show that if F is the rational numbers or a multiquadratic number field, p
is 2,3, or 5, and K/F is a Galois extension of degree a power of p, then for
elliptic curves E/Q ordered by height, the average dimension of the p-Selmer
groups of E/K is bounded. In particular, this provides a bound for the average
K-rank of elliptic curves E/Q for such K. Additionally, we give bounds for
certain representation-theoretic invariants of Mordell--Weil groups over Galois
extensions of such F.
  The central result is: for each finite Galois extension K/F of number fields
and prime number p, as E/Q varies, the difference in dimension between the
Galois fixed space in the p-Selmer group of E/K and the p-Selmer group of E/F
has bounded average.
