Abstract Algebra Answers

M3[ Z ]has no nilpotent elements.

Obtain the order of each element of ℘(S) where S={1,2,3}

Let G be a group and H be a non-empty finite subset of .G If ab∈H∀ b,a ∈ H then show that H ≤ .G Will the result remain true if H is not finite? Give reasons for your answer.

Consider the ideal I<x³-1,2x⁴+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.

Check whether or not <(x+1)²> is a maximal ideal of Z[x].

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