In Q[X] the polynomials 1 + X +
2X2 - 3X5
and 1 + X - X4 are congruent modulo X2,
because the difference 2X2 + X4 -
3X5 of the polynomials is a multiple
of X2. Indeed, 2X2 +
X4 - 3X5 =
X2(2 + X2 - 3X3).
If d is a nonconstant polynomial in R[X], where
R is a field, and if f is an arbitrary polynomial in
R[X], then dividing f by d leaves a remainder
r of degree less than deg(d). The polynomials
f and r are congruent modulo d. To see this
note that division by d produces a relation of the form
f = qd + r.
Rewriting this as f - r = qd then shows
that f and r are congruent modulo d.
In Z/5Z[X]
the polynomials 1 + X + X2 and
1 - X + 3X3 are not congruent modulo
X2, since their difference is not divisible by
X2.
In Q[X], X4 - 4
and 0 are congruent modulo X2 - 2, because X2 - 2
divides X4 - 4.
In Z/3Z[X] the polynomials X3 + 2
and 1 are congruent modulo X + 1, because