Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 4 / 12F / iv

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- Solution iv

Solution

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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