 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/ib/paper-4/12f/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

In the quotient ring, the classes of

x - 1

and

x^{2} + x + 1

are nonzero: neither nonzero polynomial of degree less than three belongs to the ideal generated by

x^{3} - 1

. But their product is the zero class, since

(x - 1) (x^{2} + x + 1) = x^{3} - 1.

The quotient thus contains zero divisors, so it is not an integral domain.

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