Advanced Ring Theory Problems (20 Marks) Discuss the relationship between integral domains, Euclidean domains, and fields. Provide examples to illustrate each. (10 Marks) Analyze the factorization properties of the ring of Eisenstein integers
Z[\omega ]
, where
\omega =e^(2\pi (i)/(3))
. Provide examples of unique factorizations in this ring. (5 Marks) Explain the significance of the characteristic of a ring using examples from matrix rings over finite fields. (5 Marks)