Question

Let \( I \) be the ideal generated by \( x^2 + x + 1 \) in the polynomial ring \( R = \mathbb{Z}_3[x] \), where \( \mathbb{Z}_3 \) denotes the ring of integers modulo 3. Then the number of units in the quotient ring \( R/I \) is \(\underline{\hspace{1cm}} \).

Hint:

In a finite field, the number of units is one less than the total number of elements. Use the degree of the polynomial to determine the size of the quotient ring.

Answer 1

Correct Answer: 6

First, observe that \( x^2 + x + 1 \) is irreducible in \( \mathbb{Z}_3 \). The quotient ring \( R/I \) is isomorphic to \( \mathbb{Z}_3[x] / \langle x^2 + x + 1 \rangle \), which is a field with 9 elements (since the degree of the polynomial is 2, and \( \mathbb{Z}_3 \) has 3 elements). In any finite field, the number of units (non-zero elements) is the total number of elements minus 1. Thus, the number of units in \( R/I \) is: \[ 9 - 1 = 8. \] Thus, the number of units in \( R/I \) is \( \boxed{8} \).