Section
4.1
Congruence modulo a polynomial
In Chapter 2 we saw how to compute with integers modulo a fixed number n. In this chapter we will do something similar, but with polynomials instead of integers. Thus we work with elements of polynomial rings R[X], with R one of the sets Z, Q, R, C and Z/nZ (with n > 1).
Let
d be a polynomial in R[X]. We define a relation on
R[X] as follows.
The polynomials a, b 
R[X] are
congruent modulo
d if there exists a
polynomial q
R[X] such that a - b = qd, in other words
if a and b differ by a multiple of d.
Notation: a = b (mod d).
Our goal will be to port as many results as possible from the arithmetic modulo an integer to the arithmetic modulo a polynomial. The following theorem tells us that, to begin with, the most important property (the division into residue classes) is preserved.
The relation `congruence modulo d' is an equivalence relation on R[X].