1.6.100 Fields, Groups, Rings

condition: Given a ring \( \mathbb{Z}[i \sqrt{3}]=\{a+b i \sqrt{3} \mid a, b \in \mathbb{Z}\} \). 1. Find all inverse elements in a given ring. 2. Find all divisors of the number 4. 3. Find all common divisors of 4 and \( 2+2 i \sqrt{3} \). 4. Prove that 4 and \( 2+2 i \sqrt{3} \) do not have a greatest common divisor.