  We study the growth of the Galois invariants of the $p$-Selmer group of an
elliptic curve in a degree $p$ Galois extension. We show that this growth is
determined by certain local cohomology groups and determine necessary and
sufficient conditions for these groups to be trivial. Under certain hypotheses
this allows us to give necessary and sufficient conditions for there to be
growth in the full $p$-Selmer group in a degree $p$ Galois extension.
