Abstract Algebra Answers

Consider the ideal I=<x³-1, 2x⁴+2x³+7x²+5x+ 5>in Q[x]

Find

p(x) belongs to Q[x]such that I=<p(x)> .

Is Q[x]/I is field ? Give reasons for your answer

Check whether or not R is a normal subgroup of the group H = (R + Ri + Rj+ Rk,+ )where i²=j²=k² =-1,ij=-ji,jk=-kj,ki=-ik=and

( a+ bi + cj+ dk) + (a ′ + b′i + ′jc + d′ )k = (a + a′) + ( b+ b′ i) + (c + c′ j) + ( d+ d′ k) for

d,c,b,a,d',c',b',a ′∈ R.

Find the number of normal subgroups of order 25, and of order 50, of a group

of order 75.

Find Z (D8) the centre of . D8 Also give the algebraic structure of D8/Z (D8.)

Define ~ on R by ‘ a ~ b iff a − b∈Z ’. Check whether or not ~ is an equivalence relation on R. If it is, find [√5 ]Else, give another equivalence relation on R

Check whether or not S={a0+a1x+.......+anx^n belongs to Z[x]|5|a0}is an ideal of Z[x]

Show that if G is a non-cyclic group of order n, then G has no element of order

n. Further, give an example, with justification, of a non-cyclic group with all its

proper subgroups being cyclic.

Let R be a ring. Show that M3(R) is a ring with respect to the usual matrix

addition and multiplication. Further, if R is commutative, will M3(R )be

commutative? Why, or why not?

Consider the ideal I<x³-1.x⁴+2x³+7x²+5x+5> in Q[x]. Find p(x) ∈Q[x] such that I =<p(x)> .

Is Q[x]/I a field? Give reasons for your answer.

Prove that R^5/R=R⁴ as rings.