IDA: Interactive Document on Algebra

Let R be a field and d a polynomial of degree n > 0 in R[X]. The residue class ring R[X]/(d) carries a vector space structure as follows.

Theorem

The residue class ring S = R[X]/(d) is a vector space of dimension n over R, with

the addition of the ring S,
scalar multiplication of the scalar r R and the vector g S given by the product rg in the ring S.

The residue classes of 1, X, ..., Xⁿ^-1 are a basis of S.