prove that any F[x]/ <r(x)>, r(x) being irreducible is a PID
and any field is a PID. perhaps what you meant to ask is: prove that if F is a field, and thus F[x] is a PID, then F[x]/<r(x)> is a field when r(x) is irreducible.
the proof might go something like r(x) irreducible --> <r(x)> is maximal, so F[x]/<r(x)> has no proper ideals.