Part CCSIR NET December 2024a-product-of-fields-is-not-a-field

A product of fields is not a field

Let $f \in R [x]$ be a product of distinct monic irreducible polynomials $P_{1}, P_{2}, \dots, P_{n}$ , where n ≥ 2. Let (f) denote the ideal generated by f in the ring $R [x]$ . Which of the following statements are true?

A. $R [x] / (f)$ is a field.
B. $R [x] / (f)$ is a finite dimensional $R -$ vector space.✓
C. $R [x] / (f)$ is a direct sum of fields, each of which is isomorphic to $R$ or $C$ .✓
D.There are no non-zero elements $u \in R [x] / (f)$ such that u^m = 0 for some m ≥ 1.✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: Every prime ideal is maximal

A product of fields is not a field

More on this topic