Exercise
Consider S = Z/11Z[X]/(X2 - a)
for a 
Z/11Z.
For which a is S a field of order 121?
2, 3, 5, 7, 9
2, 6, 7, 8, 10
1, 2, 3, 5, 7
No.
As X2 - 9 = (X - 3)(X + 3),
for a = 9, the ring S is not a field.
Yes. These values of a are the nonsquares of
Z/11Z; for these values, X2-a is irreducible.
No.
As X2 - 1 = (X - 1)(X + 1),
for a = 1, the ring S is not a field.