(too long for a comment.)
The character table of GL(2,p) in characteristic 0 can be found for example the book "A Journey through representation theory" by Gruson and Serganova on page 147.
seems to be true, but I do not know a reference (maybe books by Gordon James?). Here is a way to test it for a given prime with MAGMA (you can input it in http://magma.maths.usyd.edu.au/calc/ ). The code also works for any other finite group to get a minimal splitting field over prime fields.
p:=7; G:=GeneralLinearGroup(2, GF(p)); SIMS := AbsolutelyIrreducibleModules(G,GF(p)); temp_field_sizes:=[]; for i in SIMS do Append(~temp_field_sizes,#BaseRing(i)); end for; MaX := Maximum(temp_field_sizes); MaX;
when the result is p again, then the prime field is a splitting field. Magma confirms that the prime field is a splitting field for all primes <20.