Skip to content
Part CCSIR NET December 2023gaussian-integers

Gaussian integers

Let R=Z[X]/(X2+1)R = \mathbb{Z}[X]/(X^{2} + 1) and ψ:Z[X]→R\psi : \mathbb{Z}[X] \to R be the natural quotient map. Which of the following statements are true?

  1. A.R is isomorphic to a subring of C\mathbb{C}.✓ Correct
  2. B.For any prime number p∈Zp \in \mathbb{Z}, the ideal generated by ψ(p)\psi(p) is a proper ideal of R.✓ Correct
  3. C.R has infinitely many prime ideals.✓ Correct
  4. D.The ideal generated by ψ(X)\psi(X) is a prime ideal in R.

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: Every prime ideal is maximal

More on this topic

The chapter behind this: Ideals, quotients and the Chinese remainder theorem — free to read

From Rings and Fields › Ideals, quotient rings, prime & maximal ideals, CRT

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

ShareWhatsAppTelegram