Explanation

Of course we have to check that a different choice of representatives leads to the same residue class for the sum (and the product).

For the sum we verify this as follows. Suppose that a and a' are both representatives of the same residue class and also that b and b' represent a single class. Then there are polynomials p and q with a - a' = pd and b - b' = qd. Addition leads to the equality

(a + b) - (a' + b') = (p + q)d.

This implies that a + b and a' + b' belong to the same residue class modulo d. Hence addition is well defined.

The check for multiplication is similar. Although the point indicating multiplication does not stand for the usual multiplication of polynomials (rather, for multiplication of residue classes), it does not harm to think so (because the result is obtained as the residue class of the ordinary multiplication of two plain polynomials).

So the elements a and b in the definition of the sum and the product play no special role: Given any representative a₁ of the class a + (d) and any representative b₁ of the class b + (d), the sum of the two classes is a₁ + b₁ + (d).

In summary, to find the sum of two classes c₁ and c₂:

• Choose a representative a₁ from the residue class c₁ and a representative a₂ from the residue class c₂.
• Add the two representatives: a₁ + a₂.
• The sum of the two classes is the residue class modulo d of a₁ + a₂.

The procedure for the product is similar.