Exercise
Consider the ring R =
Z/7Z[X]/(X2-a)
where a 
Z/7Z. For which
values of a is R a field?
a = 0, 1, 3
a = 3, 5, 6
a = 2, 3, 5
No. If a = 0, then X is a zero divisor as
X2 = 0.
Correct. For these values X2 - a is irreducible.
No; 2 is a square in Z/7Z,
so X2 - 2 is reducible.