Question

Let \( \mathbb{R}[X] \) denote the ring of polynomials in \( X \) with real coefficients. Then, the quotient ring \( \mathbb{R}[X]/(X^4 + 4) \) is

• A. a field
• B. an integral domain, but not a field
• C. not an integral domain, but has 0 as the only nilpotent element
• D. a ring which contains non-zero nilpotent elements

Hint:

When analyzing quotient rings, check the factorization of the generator polynomial to determine whether the ring is a field, integral domain, or contains zero divisors.

Answer 1

The Correct Option is C

We are given the quotient ring \( \mathbb{R}[X]/(X^4 + 4) \). We need to determine the structure of this quotient ring. Step 1: Analyze the polynomial \( X^4 + 4 \)
The given polynomial is \( X^4 + 4 \), which is a degree 4 polynomial with no real roots. This polynomial is not irreducible over \( \mathbb{R} \), because we can factor it as: \[ X^4 + 4 = (X^2 + 2i)(X^2 - 2i), \] where \( i \) is the imaginary unit. Therefore, the quotient ring \( \mathbb{R}[X]/(X^4 + 4) \) is not a field since the ideal \( (X^4 + 4) \) is not maximal. Step 2: Check if it's an integral domain
For the ring to be an integral domain, it should have no zero divisors. However, since \( X^4 + 4 \) factors into nontrivial factors, the quotient ring will have zero divisors and thus is not an integral domain. Step 3: Analyze nilpotent elements
In this quotient ring, the only nilpotent element is \( 0 \), as the structure does not admit non-zero nilpotent elements (elements \( x \) such that \( x^n = 0 \) for some \( n>0 \)). Step 4: Conclusion
Thus, the quotient ring is not an integral domain, but it has 0 as the only nilpotent element.