Abstract Algebra Answers

Use fundamental theorem of homomorphism to prove that the ring R^2 and R^4/R^2 are isomorphic.

Write down all elements of quotient group Z18/<6>. Is any element of order 5?

does the ring Z2[x]/<(x^8)+1> have nilpotent elements? justify.

If F is a field with 49 elements, prove that x^49=x. for all x belongs to F. also find characteristic of F.

Does the ring Z7[x]/<x^2+3> have nilpotent elements? justify.

prove that Q[x]/<x-2> is isomorphic to Q as fields.

Is the ideal generated by x^2+1 in Z2[x] a prime ideal of Z2[x]? give reason.

consider the ideal I=12Z of Z. Find a proper ideal J of Z such that I+J=Z

Use the fundamental theorem of homomorphism to prove that rings R² and R⁴/R² are isomorphic.

(a) Let M and T be a groups and φ : M −→ T an epimorphism. Prove that if M is Abelian then T is Abelian.
(b) Let G be a group and let H be a subgroup of index 2. Prove that for every g ∈ G, g−1 / ∈ H whenever g / ∈ H. Hence, prove that for all a,b ∈ G, if a / ∈ H and b / ∈ H then ab ∈ H.
(c) Find all the distinct left cosets of the subgroup H = h(1,1)i in Z2 ×Z4.