Part CCSIR NET June 2024irreducible-quotient-need-not-be-a-ufd

Irreducible quotient need not be a ufd

For two indeterminates x, y, let R = 𝔽 $_{3} [x]$ and S = R[y]. Which of the following statements are true?

A.S is a principal ideal domain
B. $S / (y^{2} + x^{2})$ is a unique factorization domain
C.S is a unique factorization domain✓
D.S/(x) is a principal ideal domain✓

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.

50 are analysed free — try those first.

Related counterexample: Irreducible over ℤ ⇒ irreducible mod every prime

More on this topic

Part B questionJune 2023
irreducible over Q not mod pDecember 2023
odd and even powers cannot cancelDecember 2024
the cyclotomic index must be primeDecember 2024

The chapter behind this: Irreducibility tests — free to read

From Rings and Fields › Polynomial rings and irreducibility tests

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