Example

Let us look in Z/3Z[X] at residue classes modulo X² + 1. According to the proposition, every residue class has its own unique representative of degree at most 1. Since there are precisely nine polynomials in Z/3Z[X] of degree at most 1, we find exactly nine residue classes:

• 0 + (X² + 1),
• 1 + (X² + 1),
• 2 + (X² + 1),
• X + (X² + 1),
• 1 + X + (X² + 1),
• 2 + X + (X² + 1),
• 2X + (X² + 1),
• 1 + 2X+ (X² + 1),
• 2 + 2X + (X² + 1).

In practice we will of course often use the short notation 1 + X instead of 1 + X + (X² + 1), etc.