IDA: Interactive Document on Algebra

The second special case to discuss is arithmetic modulo the constant polynomial n (> 0) in the polynomial ring Z[X]. Two polynomials in Z[X] are congruent modulo n if and only if for each i, the coefficients of Xⁱ differ by a multiple of n. Therefore, each residue class has a representative whose coefficients all lie in {0, 1 ..., n-1}. This is similar to polynomials in (Z/nZ)[X]. The relation is clarified by the following map.

I : Z[X]/(n) -> (Z/nZ)[X],

a₀ + a₁X + ··· + a_mX^m + (n) -> a₀ + a₁ X + ··· + a_mX^m.

Since this map is constructed using representatives, we have to check that the result does not depend on the representatives chosen.

Theorem

The map I is well defined and has the following properties.

It is a bijection.
It respects addition: I(a + b) = I(a) + I(b).
It respects the zeros: I(0 + (n)) = 0.
It respects multiplication: I(a · b) = I(a) · I(b).
It respects the units: I(1 + (n)) = 1.

The conclusion is that the arithmetic in Z[X]/(n) is nothing but the arithmetic in (Z/nZ)[X]. In mathematical jargon: The two arithmetical structures are isomorphic (i.e., equal of form).