Question

Let \( \mathbb{Z} \) denote the ring of integers. Consider the subring \[ R = \{ a + b\sqrt{-17} : a, b \in \mathbb{Z} \} \] of the field \( \mathbb{C} \) of complex numbers. Consider the following statements: P: \( 2 + \sqrt{-17} \) is an irreducible element.
Q: \( 2 + \sqrt{-17} \) is a prime element. Then:

• A. both P and Q are TRUE
• B. P is TRUE and Q is FALSE
• C. P is FALSE and Q is TRUE
• D. both P and Q are FALSE

Hint:

In algebraic number fields, an irreducible element is not necessarily prime. Primality requires a stricter condition than irreducibility.

Answer 1

The Correct Option is B

To determine the truth of the statements, we first need to examine the properties of \( 2 + \sqrt{-17} \).

Step 1: Analyze statement P (Irreducibility).
An element is irreducible in a ring if it cannot be factored into a product of two non-units in the ring. \( 2 + \sqrt{-17} \) is irreducible because it cannot be factored into simpler elements within \( R \). Therefore, statement P is TRUE.

Step 2: Analyze statement Q (Primality).
An element is prime in a ring if it divides the product of two elements implies that it divides at least one of them. \( 2 + \sqrt{-17} \) is irreducible, but it is not prime in this ring. Therefore, statement Q is FALSE.

Final Answer: (B) P is TRUE and Q is FALSE