Which of the following is FALSE? Every field is an integral domain. Every…

1. Every field is an integral domain (True)

• A field has no zero divisors and is commutative with unity.

• Hence, every field satisfies the definition of an integral domain.

2. Every integral domain is a field (False)

• An integral domain does not require multiplicative inverses for all non-zero elements.

• Counterexample:
  Z\mathbb{Z}Z (integers) is an integral domain but not a field (e.g., 2 has no inverse in Z\mathbb{Z}Z).

3. Finite integral domain is a field (True)

• In a finite integral domain, every non-zero element must have an inverse.

• Hence, it becomes a field.

4. Zp\mathbb{Z}_pZp​ is a field iff ppp is prime (True)

• If ppp is prime → no zero divisors → field.

• If ppp is composite → zero divisors exist → not a field.