Exercise
Let R = Z/2Z and d = 1+X+X3. The following polynomial is in the residue class 1+X2+dR[X]:
X + (1 + X2)2.
(1 + X + X2)(X + X2).
(1 + X)3.
X(1 + X2).
This is true, because X + (1 + X2)2 = 1 + X + X4 = 1 + X2 modulo d.
No, because (1 + X + X2)(X + X2) = X + X4 = X2 modulo d.
But X2 and 1 + X2 are clearly not
congruent modulo d.
No, because
(1 + X)3 =
1 + X + X2 + X3 =
X2 modulo d.
But X2 and 1 + X2 are clearly not
congruent modulo d.
No, because X(1 + X2) = X + X3 = 1 modulo d.
But 1 and 1 + X2 are clearly not
congruent modulo d.