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Locate and name the singularities of the following functions in the finite z-plane:

1. ln(z+3i)/z^2

2. z^2-2z/(z^2+2z+2)

1. ln(z+3i)/z^2

2. z^2-2z/(z^2+2z+2)

If ω be the imaginary cube root of unity show that the set {1,w,w^2)is a cyclic group of order 3 with respect to multiplication.

Show that (2,X) is maximal in Z+XQ[X].

Verify that K[[X]] is a local ring,where K is a field

b) Let G be a group, H∆ G and β≤ G/H . Let A ={x∈G | Hx∈β}. Show that

i) A≤G , ii) H∆ A , iii) β= A/H

i) A≤G , ii) H∆ A , iii) β= A/H

Let R be an integral Domain then deg(fg)=degf+degg

R be a commutative ring with identity iff R[[X]] is commutative ring with identity

Is the polynomial ring R[X] a subring of R[[X]]?

let R be a commutative Ring with identity a formal power series f(X) is invertible in R[[x]] if and only if the constant term f0 has an inverse in R

Prove that R ^n/R^m~=R^n-m as groups, where n, m belongs to N such that n>=m.