73 has a root in is an irreducible pol. over and it thus splits in linear factors over (why?).
We thus have the fields tower , with .
Use now the Fund. Theorem of Galois Theory with the group of automorphisms of that leave fixed...
Of course, you STILL have to prove that the resulting group is the non-abelian group of order 6, but if you succeed with the above I think this already will be easier.