In other words, q(x) is just an irreducible quadratic polynomial over F_p. There's always at least one such polynomial because there're p^2 monic quadratics, but only p+p(p-1)/2=p(p+1)/2 are reducible, then there must be p(p-1)/2 irreducible ones
Describe how a polynomial q(x) in Zp[x] should be chosen to guarantee that :
a) ring Zp[x]/q(x)) has p^2 elements.
b) the ring Zp[x]/q(x)) is a field.
Why is the choice of such a q(x) always a possibility?
I'm not really sure how to do this