(too long for a comment.) 2. 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/ )
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.