Part CCSIR NET December 2024the-cyclotomic-index-must-be-prime

The cyclotomic index must be prime

Which of the following quotient rings are fields?

A. $Q [x] / (x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1)$ ✓
B. $Q [x] / (x^{5} + x^{4} + x^{3} + x^{2} + x + 1)$
C. $Z [x] / (x - 101)$
D. $Q [x] / (x^{43} - 41 x + 41)$ ✓

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
irreducible quotient need not be a ufdJune 2024

The chapter behind this: Irreducibility tests — free to read

From Rings and Fields › Polynomial rings and irreducibility tests

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