IDA: Example

Examples

In Q[X], the residue class modulo X containing 2 consists of all polynomials of the form 2 + Xf for some polynomial f. In other words, it consists of all polynomials whose constant term is 2.
The equivalence class contains the polynomial 2 + X + X¹⁰. So
{2 + X + X¹⁰ + Xf | f an arbitrary polynomial}
describes the same equivalence class.
Let d Z/2Z[X] denote the polynomial X² + 1. By division with remainder we can find, for each equivalence class in Z/2Z[X]/(d), a representative of degree at most 1.
Consider the equivalence class of X³. Division by X² + 1 leads to X³ = X(X² + 1) + X; in other words, X³ and X differ by a multiple of X² + 1. Hence X³ is congruent to X modulo X² + 1.
The residue class ring Z/2Z[X]/(X²) consists of precisely four classes:
- 0 + (X²),
- 1 + (X²),
- X + (X²),
- 1 + X + (X²).
To verify this, note that every class has a representative of degree at most 1 (start with an arbitrary representative, divide by X² and take the remainder), so there are at most four classes, because there are exactly four polynomials of degree at most 1.
Finally, two distinct polynomials of degree at most 1 can never be congruent modulo the degree 2 polynomial X².