Exercise
Which of the following ideals of Z/2Z[X,Y] is maximal?
({X3 - 1,Y - 1})
({X3 + X + 1,Y - 1})
({X3 + X + 1,Y2 - 1})
No.
X3 - 1 = (X - 1)(X2 + X + 1),
so the ideal is not prime, hence not maximal.
Yes. Computing "modulo Y - 1", in a sense to be made precise later,
we only need to consider X3 + X + 1 in Z/2Z[X].
This polynomial is irreducible, and so the corresponding ideal is maximal.
No.
X2 - 1 = (X - 1)2,
while X - 1 does not belong to the ideal.
Thus, the ideal is not prime, whence not maximal.