IDA: Proof

Proof

The proofs of arithmetical rules for computing modulo a polynomial follow from the corresponding arithmetical rules for addition and multiplication of polynomials.

Let a', b', c' be representatives in R[X] of the classes a, b, c in R[X]/(d).

a + b = b + c	a · b = b · a
(a + b) + c = a + (b + c)	(a · b) · c = a · (b · c)
a · (b + c) = a · b + a · c

Look at the proof of the last rule and copy...

First we look at the left-hand side of the rule to be proved. According to the definition of addition,

b + c = b' + c' + dR[X].

By the definition of multiplication, we then have a(b + c) = a'(b' + c') + dR[X]. Since a'(b' + c) = a'b' + a'c' (this is the associativity rule for polynomials), we find

a(b + c) = a'b' + a'c' + dR[X].

For the two products a · b and a · c occurring in the right-hand side of the rule to be proved, we have: a · b = a'b' + dR[X] and a · c = a'c' + dR[X]. Hence

a · b + a · c = (a'b' + dR[X]) + (a'c' + dR[X]) = a'b' + a'c' + dR[X].

Since we have found the same expression for left-hand and right-hand side, the rule is proved.

(In the proof we consistently worked with a fixed representative from a, etc.; strictly taken, this is not needed, although it turns out to be convenient.)