Abstract Algebra Answers

Let G be the set of all 3×3 non-singular matrices with entries from Q. Let X be a fixed matrix in G.prove that operation *defined by A*B= X^(-1)ABX be a binary operation. How do you prove that G is a group with respect to binary operation defined by A*B=X^(-1) ABX?

Give a non zero function from M3 (Z) to Z.

For the Ring, R= Z2[x]/<x^8-1>. Find zero divisors and nilpotent elements if any.

Using Fundamental theorem of homomorphism to prove that the rings R^2 and R^2/R^4 are isomorphic.

Give an example with justification of an element of M3 (Z) that is a unit but not the identity element.

Let G be a group and H be a non empty finite subset of G.If ab belongs to H for all a,b prove that H is a subgroup of G.will the result be true if H is not finite?

Check whether x^5+9x^4+12x^2+6 is reducible over Q.

Let S = { a + ib / a , b ∈ Z ,b is even } Show that S is a
subring of Z[i] , but not an ideal of Z[i]

Show that polynomial x⁵=9x+3 Is not solvable by radicals over.

Check whether x^5+9x^4+12x^2+6 is reducible over Q.