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

The cyclotomic index must be prime

Which of the following quotient rings are fields?

  1. A.Q[x]/(x6+x5+x4+x3+x2+x+1)\mathbb{Q}[x]/(x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1)✓ Correct
  2. B.Q[x]/(x5+x4+x3+x2+x+1)\mathbb{Q}[x]/(x^{5} + x^{4} + x^{3} + x^{2} + x + 1)
  3. C.Z[x]/(x−101)\mathbb{Z}[x]/(x - 101)
  4. D.Q[x]/(x43−41x+41)\mathbb{Q}[x]/(x^{43} - 41x + 41)✓ Correct

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, and why this one tests standard counterexample.

See pricing

50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: Irreducible over ℤ ⇒ irreducible mod every prime

More on this topic

The chapter behind this: Irreducibility tests — free to read

From Rings and Fields › Polynomial rings and irreducibility tests

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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