Ok, let be one of those monic quadratics which is also irreducible ( can you prove such a thing must always exist? Note that according to what
you said you can!)) , so that the ideal is maximal in (why?) the factor ring is a field (why?) .
Well, now just prove that ...
Hint: every element in as defined above has the form , but using Euclides algorithm in show that any element in has a
representative with and any two of them is a different element in the field.
Well, now just count up all the different constant and linear poilynomials in ...