Algebra - Mathematics Previous Year Questions

The value of \( a \) is

Consider the following statements: I. The ring \(\mathbb{Z}[\sqrt{-1}]\) is a unique factorization domain. II. The ring \(\mathbb{Z}[\sqrt{-5}]\) is a principal ideal domain. III. In the polynomial ring \(\mathbb{Z}_3[x]\), the ideal generated by \(x^3 + x + 1\) is a maximal ideal. IV. In the polynomial ring \(\mathbb{Z}_2[x]\), the ideal generated by \(x^3 + 1\) is a prime ideal. ( \(\mathbb{Z}\) denotes the set of all integers, \(\mathbb{Z}_n\) denotes the set of all integers modulo \(n\), for any positive integer \(n\) ) Which of the above statements are TRUE?

Let \(G\) be a group of order \(5^4\) with center having \(5^2\) elements. Then the number of conjugacy classes in \(G\) is ________.